enigMania

Finished

2012-01-06T17:02:00.004Z

Being nearly there

2011-07-31T16:28:00.006+01:00

The summer holidays being here, consider Zeno taking his family on a trip. One of his kids asks ‘Are we nearly there?’ Suppose the answer is ‘No’, as it will be for most of their journey. Typically, the question is repeated a short time later. Since they will not have moved very far in that short time, how could they be nearly there? A bit further on from any place that was not nearly there would still not be nearly there. The Sorites paradox is that it follows logically that they will never be nearly there. But as Zeno knows, children are impossible.

Eternity revisited

2011-07-31T16:20:00.003+01:00

The current version of my reply to Mawson is now Eternity revisited.

The Liar Paradox

2011-06-02T08:49:00.007+01:00

Over two and a half thousand years ago, the Cretan Epimenides called all Cretans liars. But while there is an air of paradox about that, liars do not always lie. So, to see the paradox more clearly, suppose that Tiberius says ‘this that I am now saying is a lie’ (without having said anything else to which he might be referring).

If what he said was true, he would have been – as he said he was – deliberately saying something false. So unless what he said was true and false, it was not true. So he did not deliberately say something false. Did he mistakenly believe that he was telling the truth? But how could he have believed that what he said was true – that he was lying – without thereby believing himself to be saying something false?

Perhaps he did not know what he was saying, but if that is the only coherent possibility, then he could not possibly have known what he was saying. And yet what he said was not nonsense. Had it been, the above would have been impossible to follow. What he said was therefore paradoxical. And to see the paradox even more clearly, consider the simpler assertion ‘this is not true’, where that ‘this’ refers to that very assertion.

If an assertion is true, then what it asserts is the case, so if ‘this is not true’ is true, then since it is self-referential, it is not true. Does that mean that it is not true? That would follow from it being either true or not (since even if it is true, it is not). The paradox is that, were it not true, its description of itself as not true would be correct. In general, if what is asserted is the case, then the assertion is true.

Assertions are true when, and only when, what they assert is the case. E.g. the description ‘snow is white’ is true if, and only if, snow is white, which clearly generalises to any description. And the self-description ‘this is not true’ is true if, and only if, it is not true. Since ‘not true’ applies when, and only when, ‘true’ does not, hence our self-description cannot be true and not true. So it cannot be true – since if it is, it is not – but what is the alternative? If it is not true, then it is true. And we cannot even conclude that it is neither true, nor not true, because that is just to say that it is not true, and true.

Perhaps the sentence ‘this is not true’ cannot coherently be interpreted as describing itself. There would be no paradox if, from that sentence failing to express a truth, it did not follow that it was a true self-description, but rather that the attempt at self-reference had failed. And we did leave Tiberius unable to know what he was saying, as though there was nothing for him to know. However, self-reference is not usually a problem, e.g. ‘this is not French’ seems true enough. And a paradox without self-reference, but otherwise very like the Liar, was introduced by Stephen Yablo in 1993.

In our version of Yablo’s paradox, Tiberius has been around forever, and until today the only claims he ever made were a rather repetitive ‘no claim made earlier by me was true’, which he said once a year throughout his infinite past. Had none of those claims been true, each would thereby have been true. But if any one of them had been true, then none of those made earlier would have been true. And in particular, the one made a year earlier would not have been true, even though none of the earlier claims would have been true.

Even so, it remains possible that the sentence ‘this is not true’ cannot coherently be interpreted as describing itself. E.g. Alfred Tarski suggested in 1935 that ‘true’ was equivocal – if not inconsistent – in such paradoxical contexts. So maybe each claim made by Tiberius gave ‘true’ a slightly different sense. Nevertheless, we intuitively take ‘true’ to be unequivocal, at least in descriptive contexts. And while there are lots of other possibilities – e.g. Graham Priest suggested in 1987 that ‘not’ allows descriptions to be true and false – common sense must make us wonder whether we are forced, by the Liar paradox, to entertain such counter-intuitive possibilities.

I shall be arguing that we are not, because there is a common-sense resolution. Our words do not describe a black-and-white world, and so truth is not an all-or-nothing affair. So self-descriptions like ‘this is not true’ are neither simply true, nor simply not true, but are rather vaguely true. My explication of that will be as simple as possible, in order to show how it is little more than common sense. And to begin with, note that colours do not divide into those that are blue and those that are not blue.

Some analytic philosophers would disagree, but it is only common sense that there is no dividing line between the blue and the other colours. Given some spectrum, the colours on either side of any such line would be indistinguishable, but colours that appear identical will both be blue enough to count as blue if one is. So there is no such dividing line. Rather, there are colours that are about as blue as not. We might call such colours ‘vaguely blue’. And if you said ‘that is blue’ of something vaguely blue, would what you said not be vaguely true (about as true as not)? Let me explain why I think that it would be.

Descriptions are true when they describe how things are, rather than how they are not, but descriptive accuracy is in general a matter of degree. E.g. ‘blue’ describes royal blue more accurately than it describes a faintly greenish turquoise. So we should say that descriptions are true when, and insofar as, they are accurate. E.g. ‘snow is white’ is true insofar as snow is white. Of course, ‘snow is white’ is usually true enough to count as simply true. But snow can also be a bit bluish, or discoloured by dirt, or sparkle with all the colours of the rainbow because it is, on closer inspection, transparent. (Whether or not snow is white therefore depends on the context of ‘snow is white’.)

Descriptions that are not true enough to count (in the given context) as true can usually be replaced with more accurate descriptions, e.g. ‘that is vaguely blue’. But we are considering particular descriptions – ‘that is blue’ and ‘this is not true’ – and wondering just how true they are. Since descriptions are true insofar as they are accurate, hence ‘that is blue’, said of something vaguely blue, is as true as not. Similarly, ‘this is not true’ is true insofar as it is not true, so it is as true as not. In other words, such descriptions are vaguely, but only vaguely, true. To see more clearly how that is only common sense, we should go more slowly through another example.

Suppose that, out of the blue, Tabitha says ‘this that I am now saying is not true’. She has said, in effect, that what she said was not a good enough description of itself for it to count – in the context of her utterance – as simply true. And what she said was nothing if not self-contradictory, so it was not describing itself very well. But therefore it seems to have been describing itself quite well.

What that shows is that self-contradictions are not always false. Usually they are, e.g. ‘this is not an assertion’ is simply false. But what Tabitha said was almost true enough to count as fairly true, falling short of that rather vague standard in order to avoid paradox. That is a coherent possibility because if what she said was vaguely true – if it is vaguely true that what she was saying was not true – then it need only follow that what she said was vaguely untrue (about as untrue as not), which clearly coheres with it being only vaguely true (about as true as not). And since all the other possibilities appear to be incoherent (or at least implausible), then that was what it was (or probably was).

What Tiberius said would also have been vaguely true, if he had known what he was saying (had he not, what he said would have been false). And for yet another variation, suppose that Tabitha knew that what she was saying was only vaguely true, so that she said ‘this that I am now saying is not true’ with the intention to say something true. Since she did not actually say ‘is vaguely untrue’, what she said would still have been only vaguely true.

What she said did seem true when we thought of it as not true, and then untrue when we thought of it as true. But that was when those two inaccurate descriptions were each creating a misleading context for the other. In fact, what she said had only the one context, that of its utterance. A nice analogy is someone wondering whether the colour of some blue-green object is really a sort of green (a bluish green). As she thinks of it as possibly green – and hence sees it, in her mind’s eye, against the various shades of green that it might be – it would probably look bluer, because the contrast would tend to enhance its bluishness. She might even wonder if it was really a sort of blue (a greenish blue). But similarly, it might thereby seem not to be, especially if it was really as blue as not.

For yet another kind of Liar paradox (with indirect self-reference), consider the following pair of sentences (read in the obvious way): The next description is true. The previous description was not true. They are paradoxical because if the first description is true then, via the second, it is not, and vice versa. But if the first is vaguely true then it follows that the second is vaguely true, and hence that the first is vaguely untrue, which coheres with it being vaguely true. Indeed, if that is the only coherent possibility – within the bounds of common sense – then those descriptions are both vaguely true.

We can hardly check all variants of the Liar paradox one by one, to see that they can all be resolved like that. But we can – and should – examine the most difficult to resolve. Suppose that ‘this is not even vaguely true’ was said, self-referentially. This new self-description seems, in effect, to have asserted its own untruth, much like the others. Yet how could it be vaguely true? Were it vaguely true, what was said – that what was said was not vaguely true – would seem false, not just vaguely untrue. Indeed, it would then seem true, since false. So this new self-description may well be hard to resolve.

We have been using ‘vaguely true’ to mean about as true as not, though. And a description that is not even vaguely true in that sense need not be completely untrue, so long as it is significantly less true than untrue. So I was equivocating when I took the new self-description to be asserting its own untruth. I was taking ‘not even vaguely true’ to mean the same as ‘not true’. For clarity, we should stick with the former sense.

It is still true that if the new self-description is vaguely true, then the assertion that it is not even vaguely true will be false. But we also know, from the previous paragraph, that if this self-description is a lot less true than untrue, then the assertion that it is not even vaguely true will be true. And if we look in between those two extremes, we find that this self-description can be a bit more vaguely true – a bit more untrue than true – while the assertion that it is not even vaguely true is also, coherently, a bit more untrue than true.

Our difficult self-description is therefore more vaguely true (less vaguely untrue). And similarly, the self-description ‘this is only vaguely true’ would seem true if vaguely true, and vaguely true if true, and is therefore less vaguely true (more vaguely untrue). That is a little complicated, so note that we might, more loosely, call either description ‘vaguely true’. To see that more clearly, let us glance at the analogous problem of higher-order vagueness.

One problem with vagueness is that we cannot, without contraction, think of the vaguely blue colours as neither blue nor (in the same context) not blue. Some philosophers therefore think of them as neither definitely blue nor definitely not blue. But then they face a problem of higher-order vagueness, the question of what happens between the definitely blue and the vaguely blue colours. We want a gap there, rather than a line at which the definitely blue looks just like the vaguely blue. But we cannot, without contradiction, think of the colours in that gap as neither definitely blue nor vaguely blue, because then they would be neither definitely blue nor (in the same context) not definitely blue.

Nevertheless, while we would usually avoid calling vaguely blue colours ‘blue’ or ‘not blue’, that is because calling them either would be only vaguely true, not because it would be false. Blue shades smoothly into green, via blue-green; and a pretty good description of the blueness of any blue-green colour might be ‘vaguely blue’, even though a better description could, for some of them (in some contexts), be ‘green’. And similarly, ‘vaguely true’ would be a fairly good description of any of the self-descriptions that give rise to Liar paradoxes (especially in view of the vagueness of ‘vaguely’), for all that it can be misleadingly inaccurate when the self-descriptions use ‘vaguely true’ themselves.

Now, as well as various variants of the Liar paradox, there are also various other paradoxes that should, intuitively, have very similar resolutions. We have already met Yablo’s paradox, which concerns an infinite set of descriptions, each asserting the untruth of all the earlier ones. And that paradox does have a common-sense resolution. If all those descriptions were vaguely true then, from any of them being vaguely true, it need only follow that all of those before it were vaguely untrue. So that is a coherent possibility; and if it is the only one (within the bounds of common sense), then the descriptions of Yablo’s paradox are (probably) vaguely true, more or less.

It appears, then, that we find such descriptions paradoxical because of a natural tendency to ignore descriptive imprecision. That tendency helps us to focus on the most apposite elements of truth and falsity in what is being said. So it is usually useful. We just have to take care with self-descriptions like ‘this is not an accurate description of itself’. Our next (and final) paradox is very similar to that Liar paradox, since it concerns the predicate expression ‘does not describe itself accurately’. (Regarding the other paradoxes of self-reference, the question of how similar they are to the Liar depends on how they should be resolved, so they would take us too far afield.)

The expression ‘is long’ is not long, not for a predicate expression. So it does not, as a rule, describe itself accurately. By contrast, ‘is short’ does. The question is, does ‘does not describe itself accurately’ describe itself accurately, or not? If it does – if it is described by ‘does not describe itself accurately’ – then it does not describe itself accurately. But therefore, since it only fails to accurately describe expressions that are describing themselves accurately, it does describe itself accurately.

To resolve this paradox we need only assume that descriptive accuracy might be a matter of degree. And for convenience, let us say that an expression is heterological when, and insofar as, it does not describe itself accurately. It follows that ‘is heterological’ is heterological insofar as it is not. So it is as heterological as not. In other words, the expression ‘does not describe itself accurately’ is vaguely heterological. And it follows that descriptive accuracy is, in general, a matter of degree. (Incidentally, Kurt Grelling and Leonard Nelson introduced the term ‘heterological’, along with this paradox, in 1908.)

This may therefore be a good place to stop, and review the common-sense resolution of the Liar (and Yablo’s) paradox. Descriptions are true when, and insofar as, they are accurate. So the self-description ‘this is not true’ is true insofar as it is not true, and so it is as true as not. Indeed, all such descriptions are vaguely true (about as true as not), more or less. That resolution is simple, and intuitive. But it is hard to find it in the literature. And because it tends to be overlooked, the reasons for its neglect are also obscure. So let me close with one possibility.

Logicians often use ‘1’ to signify truth, and ‘0’ for falsity, and the so-called fuzzy logicians use the number ½ to model half-truths. Fuzzy logic developed out of fuzzy set theory, and the most influential paradoxes of self-reference were those of set theory, as axiomatic set theory became the standard foundation of mathematics. Fuzzy logic is a mathematical logic, not common sense, but the former tends to be more attractive to analytic philosophers, who may well have preferred the precision of ½ to such vague words as ‘vaguely true’. Nevertheless, our words are unlikely to be much better defined than our purposes have required them to be, and even formal terms must ultimately derive their meanings from natural language.

The Irony Age continued

2011-04-06T09:29:00.003+01:00

My previous post was quite brief, so here are a few more thoughts on the two paragraphs quoted therein. The first paragraph was about estimating the danger posed by the LHC to the planet (if not the universe), and the big argument for safety is that cosmic rays produce such collisions all the time. Whatever a collider might produce, it’s very likely that such has already been produced on the moon, for example, lots and lots of times. And of course, the moon’s still there. That argument doesn’t seem to depend upon the niceties of particle physics. But cosmic rays spread out from the sun. So they are most concentrated near the sun. What if some merging of products of collisions is most likely nearest the sun? Such events might not occur on the moon, but might occur in the most concentrated beams of our biggest colliders. So how likely is it that such an event causes tiny ripples on the sun? The problem is that such an event would destroy the earth. And our most popular theories have little to say about such questions (and did fail to predict dark matter).

Furthermore, suppose we could estimate the answer at no more than one in a billion. Would that be safe? We are talking about the possible destruction, not only of less than ten billion people, but of all possible future human beings. What figure should be given to that? So we also need some way of determining just how safe a one-in-a-billion chance of destroying the human race really is (as Sample noted). Since that problem is so intractable (cf. the St. Petersburg Paradox), surely the main thing here is that we do have better things to do, things associated with more mundane risks (and more immediate benefits). Even theoretical physicists have plenty of other puzzles to solve. A competitive Academia may encourage them to excel at the language game of string theory, but surely the most puzzling thing in theoretical physics (given the materialism there) is the absence of anything at the fundamental level that could conceivably give rise to awareness when the fundamental particles are parts of complicated biochemical systems (the elephant in the room in which we debate synthetic biology).

Or they could address their big methodological problem, which is the question of what they should be thinking they’re doing. The language of science is mathematics, but standard mathematics is heavily influenced by Formalism, which encourages the move from general interest in a new type of theory, to the adoption of the presuppositions of that type of theory. We all know what is meant by ‘1 + 1 = 2’, but standard mathematicians will tell you that it means that {0, {0}} follows {0} in the von Neumann series (where those are ZFC sets, and ‘0’ denotes the empty set). They will say that that is just the language game that is modern mathematics. But mathematics is not a game, but the language of science; and that word ‘language’ is being used metaphorically. The literal languages of science are our natural languages, which include mathematical terminology when one is doing science. It is a philosophical question, what those terms refer to, if anything; but the meaning of mathematical statements is clearly akin to logic (not a made-up, formal logic, but the logic that all scientists should apply).

The Irony Age

2011-04-05T17:57:00.006+01:00

One problem that continues to plague discussions over the safety of particle colliders, though the issue is relevant to other areas of cutting-edge science, such as synthetic biology and genetics, is that it is impossible to have what could sensibly be called an informed public debate on the issues. The people who understand the issues best work in the field under debate, so the accusation of vested interests cannot be avoided. Ironically, the most high-profile opponents to a new technology are often so badly informed they are quickly dismissed as crackpots, and rightly so. The result is an illusion of public debate. Ill-informed opponents do a disservice to people with genuine interest and concern by squandering the opportunity for an even-handed discussion of the risks.

That paragraph is from p.193 of Ian Sample’s Massive (Virgin Books, 2011), and it reminded me of the following, from the bottom of p. 640 of R. W. Hamming’s ‘Mathematics on a Distant Planet’, American Mathematical Monthly 105 (1998), 640–650, the gist of which was that we should have based our mathematics on known truths, not—as was increasingly done throughout the twentieth century—on axioms taken to lie beyond truth and falsity.

I need to mention a few things in my life that have shaped my opinions. The first occurred at Los Alamos during WWII when we were designing atomic bombs. Shortly before the first field test (you realise that no small scale experiment can be done—either you have a critical mass or you do not), a man asked me to check some arithmetic he had done, and I agreed, thinking to fob it off on some subordinate. When I asked what it was, he said, “It is the probability that the test bomb will ignite the whole atmosphere.” I decided I would check it myself! The next day when he came for the answers I remarked to him, “The arithmetic was apparently correct but I do not know about the formulas for the capture cross sections for oxygen and nitrogen—after all, there could be no experiments at the needed energy levels.” He replied, like a physicist talking to a mathematician, that he wanted me to check the arithmetic not the physics, and left. I said to myself, “What have you done, Hamming, you are involved in risking all of life that is known in the Universe, and you do not know much of an essential part?” I was pacing up and down the corridor when a friend asked me what was bothering me. I told him. His reply was, “Never mind, Hamming, no one will ever blame you.” Yes, we risked all the life we knew of in the known universe on some mathematics. Mathematics is not merely an idle art form, it is an essential part of our society.

Academics are of course free to pursue whatever interests them; and a hundred years ago, set theory interested many pure mathematicians. But academics will only be successful if their interests are those of their peers (or industry), so there’s some irony there. Any young mathematician bothered by set theory would have been unlikely to have gone into pure mathematics. Perhaps physicists uninterested in string theory are unlikely to choose theoretical physics; but certainly, a slight bias can become the rule, over a century or so. And a rule will tend to exclude other possibilities (even in the absence of corruption), making it impossible to estimate costs and benefits properly. (To be continued.)

Russell, Cantor, Curry, and Simmons’ paradox

2011-04-04T00:01:00.001+01:00

This post is the fourth part of Vaguely True Liars.

The paradoxes of infinity being tangential here, I’ll just touch upon Russell’s paradox, which concerns collections (although it was originally formulated in terms of predicates). The obvious collections – e.g. the class of all chairs – don’t include themselves, as members (as one of the collected things), but some might, e.g. the collection of all the non-chairs would not itself be a chair. So, let the collection of all the non-self-membered collections be called ‘R’. One member of R is the class of all chairs; but is R a member of itself, or not? If it is then it’s self-membered, so it shouldn’t be; but if it isn’t then it’s eligible to be, so it ought to be. That’s Russell’s paradox. The obvious resolution – akin to my first suggestion for Grelling’s paradox – is to consider, not R, but the collection of all the other non-self-membered collections.

If I’m right about Grelling’s paradox, predicates won’t always be associated with classes, so it would be reasonable to resolve these two paradoxes differently. But of course, things are not quite that simple. We should, for example, replace ‘non-self-membered’ with ‘well-founded’ if we want to rule out such possibilities as a collection V that contains U and all the other non-self-membered collections, where U contains V and all those others. And then there’s Cantor’s paradox: If there was a (well-founded) collection of all the other (well-founded) collections, then there would be more sub-collections than there are collections, via Cantor’s diagonal argument, which shows that any collection – even an infinitely big one – has more sub-collections than it has members (at least if it’s a well-founded and non-variable collection) [i]. However, a sub-collection (of some collection) is just another collection (of some of the original collection’s members), and so Cantor’s argument is that any collection of all the others would be bigger than itself. For such reasons, Russell’s paradox may well belong with the paradoxes of infinity [ii].

So let’s look next at a paradox due, in its essentials, to Haskell Curry. Suppose I say ‘if what I’m now saying is true, then pigs fly’. That’s clearly making the same type of assertion as my earlier Liar statement. But if we generalise ‘pigs fly’ to ‘the Pope’s next assertion is true’, we get the following argument for papal infallibility. Suppose I say ‘if what I’m now saying is true, then the Pope’s next assertion is true’. And suppose, just suppose, that what I said was true. What I said was that if what I said was true, then so was whatever the Pope then said. So we are supposing that if we suppose that what I said was true – as we are doing – then we are also supposing that what the Pope said next was true. So we are, in effect, supposing (just supposing) that what the Pope said was true. To recap, we supposed that what I said was true, and thereby supposed that what the Pope said was true. But that means that what I said – that if what I said was true, then so was what the Pope said – was true, and hence that what the Pope said next was true.

But of course, the Pope might, for all we know, have said that pigs fly, so the argument was certainly fallacious. It would be interesting to consider why it was; but what’s most apposite here is how unlike the Liar paradox it was. The most obvious difference is that the word ‘not’ did not appear in it (although ‘not’ could be introduced via the material conditional) [iii]. Furthermore, by deriving fact from mere possibility, Curry’s paradox can seem more like the Ontological Argument, than the Liar paradox. Now, I suppose that how we resolve my particular example should depend upon how we treat future contingents, as well as how we treat self-reference. Regarding the former, note that what I said would have been like my Liar statement (or possibly true), had the Pope next said something false (respectively true).

But in view of the latter, Simmons’ paradox (which I recently considered in the post that that link links to) is more apposite, because that paradox can be resolved by noting that reference is in general a matter of degree. So to sum up, it may well be that a natural kind of paradox arises from our tendency to ignore – to see past – the imprecision of description. As we use language, we naturally focus upon apposite elements of truth and falsity – much as we see the picture, not the pixels – and so we tend to overlook such possibilities as Liars being vaguely true. An obvious danger is that, by considering too few possibilities, we misconstrue the evidence for our favourite logic. Indeed, since standard set theory was developed in response to such paradoxes of infinity as Cantor’s, there’s some danger of the language of modern science having been built on shaky foundations [iv].

[i] For the mathematical details, see any introduction to set theory. For a mathematician’s view of the philosophical details, see Peter Fletcher, ‘Infinity’, in Dale Jacquette (ed.), Philosophy of Logic (Amsterdam: Elsevier, 2007), 523–585.

[ii] For more on the kind of paradox that includes Russell’s and Cantor’s (and the Burali-Forti), see Stewart Shapiro and Crispin Wright, ‘All Things Indefinitely Extensible’, in Agustin Rayo and Gabriel Uzquiano (eds.), Absolute Generality (Oxford: Clarendon Press, 2006), 255–304.

[iii] For some discussion, see Graham Priest, Beyond the Limits of Thought, second edition (Oxford: Clarendon Press, 2002), 168–9, 278.

[iv] For historical details, see Ivor Grattan-Guinness, The Search for Mathematical Roots, 1870–1940: Logics, Set Theories and the Foundations of Mathematics from Cantor through Russell to Gödel (Princeton University Press, 2000).

My ‘revenge’ paradox (and Yablo’s paradox)

2011-04-03T03:32:00.004+01:00

This post is the third part of Vaguely True Liars.

In my previous post I resolved a Liar utterance; can something similar be said of the other Liar sentences? Since there are many such sentences, let’s just see if we can resolve the trickiest. In general, the most difficult sentences for any putative resolution are those that threaten so-called ‘revenge’ paradoxes, which is when the terminology of the resolution – in our case, ‘vaguely’ – is used to make a new Liar sentence that resists resolution along the same lines. So suppose I said ‘what I’m now saying is not even vaguely true’. Something that’s not even vaguely true is, prima facie, something untrue. And if this new statement is asserting its own untruth then, like my previous utterance, it should be vaguely true. But then what I said – that it wasn’t vaguely true – would seem to have been false, not just vaguely untrue. And it would then, if false, seem to be true, rather paradoxically. So this does seem to be my ‘revenge’ statement.

Was it asserting its own untruth? Something that’s not even vaguely true (in our sense) is something that’s at least a little less true than not. And that’s compatible with it being more vaguely true, in the sense of more roughly as true as not. (It’s also compatible with it being false, of course.) Now, because of that compatibility, my ‘revenge’ statement seems true if more vaguely true, as well as false if vaguely true. Is there something in between the vaguely and the more vaguely true? Well, what about the vaguely more vaguely true? That’s a clumsy turn of phrase, but it is describing an unusual statement. And perhaps we could more loosely say that my statement was vaguely true, and then qualify that by adding that, more precisely, it’s vaguely more vaguely true than the vaguely true it refers to. (How vague the latter was would depend upon what counted as true when I uttered my ‘revenge’ statement.)

That’s a bit obscure, but such obscurity would at least explain how this resolution could have been overlooked, were it correct. And we can get some clarification by glancing at the related problem of higher-order vagueness [i]. That problem might, for example, arise with vaguely bluish colours – those that are roughly as blue as not (in some context) – were we tempted to regard them as neither blue nor not blue (in that context). If they weren’t blue and yet were blue, in the same context, then they would be contradictory, so we should resist that temptation. However, being so tempted we might take them to be, for example, neither definitely blue nor definitely not blue. And then we would face a ‘revenge’ problem, via the question of what happens between the definitely blue and the vaguely bluish colours. Were these colours neither definitely blue nor vaguely bluish, and the latter not definitely blue (nor definitely not blue), then these colours would be neither definitely blue nor not definitely blue, in the same context.

That’s a problem of higher-order vagueness. But such problems don’t affect the common-sense approach that I’ve been taking. While I don’t want to call vaguely bluish colours ‘blue’ or ‘not blue’, that’s only because calling them either would be only vaguely true, not because it would be false. Blue shades smoothly into blue-green, in reality. And a colour that’s roughly as blue-green as it is blue might be described quite accurately as ‘blue’ (since it’s a faintly greenish blue). Similarly, a good description of my ‘revenge’ statement could be ‘vaguely true’ (qualified as required). After all, the term ‘vaguely’ is an especially imprecise term, and highly context-sensitive, being generally used to gesture beyond the other adjectives in use, or made apposite by such use.

Things can be made more complicated, of course. E.g. a more awkward ‘revenge’ paradox might be based on Yablo’s paradox [ii]. Imagine a place where ‘no previous utterance here was even vaguely true’ was said by someone, once a year – every year, throughout the infinite past – with nothing else ever being said there. Now, it seems to me that all those utterances would have been (vaguely more) vaguely true. But it would certainly be more awkward to justify that view in this case. Still, suppose my approach failed here; that might only mean that this paradox was more like the paradoxes of infinity than the Liar. Certainly, this paradox concerns an infinite set of semantically ungrounded sentences, rather than self-description (or circular description [iii]). And the paradoxes of infinity are not our topic. (To be continued.)

[i] That problems of higher-order vagueness are akin to ‘revenge’ paradoxes was noted by Mark Colyvan, ‘Vagueness and Truth’, in Heather Dyke (ed.), From Truth to Reality: New Essays in Logic and Metaphysics (Abingdon: Routledge, 2009), 29–42.

[ii] Stephen Yablo, ‘Paradox without Self-Reference’, Analysis 53 (1993), 251–2.

[iii] A simple example of a circular Liar is the following pair of sentences. The next sentence is true. The previous sentence was false. I regard them both as vaguely true, when read or said in the obvious way.

Liar statements are about as true as not

2011-04-02T13:26:00.002+01:00

This post is the second part of Vaguely True Liars.

Things are usually described well enough, for some obvious purpose. E.g., if something is obviously blue, then we might call it ‘blue’ when referring to it. If, given a different object, some other description sprang to mind, well, maybe the object wasn’t blue. Or perhaps it was blue, but something else about it was more apposite. Another possibility is that it was as blue as not, however, because colours don’t divide into those that are blue and those that aren’t. On the two sides of any such line, between the blue and the other colours of some spectrum, would be colours that were indistinguishable; but of course, colours that appear identical will both be blue enough to count as blue if one is. So there’s no such dividing line; rather, there are colours that are vaguely bluish. Intuitively, ‘that’s blue’ said of such colours would be vaguely true. It would not be true enough to count as true, but being roughly as true as not, nor would it be more than vaguely false.

There are two basic logical possibilities, i.e. true, or not. Statements are true insofar as they describe how things are, as opposed to how they aren’t. And when a description isn’t true enough, we can usually replace it with a more detailed description. E.g. we can replace ‘that’s blue’, when it’s vaguely true, with ‘that’s vaguely bluish’. And we don’t always have to make things so explicit, because we naturally focus upon the pertinent elements of truth in what’s being said (or perhaps upon some obvious falsity). Indeed, that may well be why we have the concept of truth (and that of negation) [i]. Perhaps it’s also why we fall for the Liar paradox.

Suppose I say ‘what I’m now saying isn’t true’. If what I said was true, then as I said, what I said wasn’t true. Does it follow that what I said wasn’t true? The paradox is that if so, then since that’s what I seem to have said, I seem to have said something true. So you may well wonder if I really said anything, with my Liar utterance. But if not, then surely you would have found my utterance incomprehensible, rather than paradoxical, and so I think that the meaning of my utterance must have been fairly clear. It seems to me that I was saying that what I was saying wasn’t a good enough description of itself for it to count as simply true. Now, since it was nothing if not self-contradictory, it wasn’t describing itself very well. But therefore it seems to have been describing itself quite well after all. Still, perhaps its self-description was almost good enough to count as simply (or absolutely) true, but its self-contradictory nature meant that it fell just short enough to avoid paradox.

Much as ‘is heterological’ had to be as heterological as not, my Liar utterance seems forced to be about as true as not. I say ‘about’ in view of the underlying imprecision of natural language (and presuming more accuracy could lead to a ‘revenge’ paradox that would take us back to this position anyway). The paradoxical reasoning rules out the non-vague extremes, but it being vaguely true that what I said was not true implies only that what I said was vaguely untrue – vaguely false (since I was making an assertion) – which coheres well enough with it being vaguely true for there to be no more contradiction. And this resolution also explains why my utterance seemed true when thought of as false, and vice versa. By analogy, if you were given something blue-green, for example, you might wonder whether it was really more green than blue. But if it’s roughly as blue as not, then it would look bluer as you postulated it amongst – and hence saw it in your mind’s eye against – various shades of green. The contrast would enhance its bluishness. And if you thence thought of it as possibly blue, it would similarly seem not to be.

You may be wondering what exactly the element of truth would have been, were my statement vaguely true. Well, it would also have been vaguely false, so it would have been vaguely true that it was false. So we might say that the element of truth was that there was an element of falsity (and vice versa) [ii]. But more precisely, I’m suggesting that my statement wasn’t describing itself very well, that it was neither true enough to count as simply true, nor sufficiently false to be less than vaguely true. It may well have seemed untrue (if true) and then true (if untrue), but that was while those two inaccurate descriptions were being each other’s context. My statement had only the one context of its utterance. And it must have been about as true as not, if the alternatives are paradoxical. (To be continued.)

[i] The more usual reason given for why we have the concept of truth is that it allows such sweeping claims as ‘everything the Pope said was true’. For more on that reason, see John Collins, ‘Compendious Assertion and Natural Language (Generalized) Quantification: A Problem for Deflationary Truth’, in Cory D. Wright and Nikolaj J. L. L. Pedersen (eds.), New Waves in Truth (Basingstoke: Palgrave Macmillan, 2010), 81–96.

[ii] Elements of truth and falsity are often propositions. But it might be argued that Liar sentences express no proposition, in their paradoxical contexts; and doubts about emphasising propositions have, for example, been raised by W. V. Quine, Philosophy of Logic (Englewood Cliffs: Prentice-Hall, 1970), 8–13.

Introduction (via Grelling’s paradox)

2011-04-01T07:44:00.004+01:00

This post is the first part of Vaguely True Liars.

To begin very simply, ‘is long’ is not long, not for a predicate expression. Kurt Grelling called it ‘heterological’. Heterological expressions don’t apply to themselves. Grelling asked, is ‘is heterological’ heterological? It is if it doesn’t apply to itself – that’s what ‘heterological’ means – but if it is, then it applies to itself – that’s what ‘applies’ means – and so it isn’t heterological. It is if it isn’t, and it isn’t if it is; that’s Grelling’s paradox [i]. Could we arbitrarily include ‘heterological’ in, or else exclude it from, the range of its application? But then ‘heterological’ would not, in that particular instance, mean what it should, intuitively, mean. Furthermore, Grelling’s paradox is of a kind with the Liar paradox (our main topic), and taking such an approach to the Liar would amount to denying the universal relevance of truth (or its coherence) [ii].

The approach I take to such self-descriptive paradoxes begins by noticing that descriptive accuracy is in general a matter of degree. I would say, for example, that because ‘is pretty’ makes us think of prettiness, it’s vaguely pretty, but only vaguely pretty because it’s only a word (outside of calligraphy or song). It’s a matter of opinion, of course; but what about ‘is a bit long’, or ‘is a little lengthy’, or ‘is only slightly lengthy’? Anyway, if descriptive accuracy is a matter of degree, in general, then I should have said that a predicate expression is heterological insofar as it doesn’t apply to itself. And if that is – or should be – what ‘heterological’ means, then ‘is heterological’ is heterological only insofar as it isn’t. That is, it’s as heterological as not. So we might say that it’s only vaguely heterological.

And descriptive accuracy is likely to be a matter of degree, in general (in various context-sensitive ways). Our words are unlikely to be much better defined than our purposes have required them to be, and so we should expect a ubiquitous – since ordinarily unobtrusive – imprecision throughout our natural languages. Such imprecision is not usually important – that’s why it’s there – and even when it does matter, we can always clarify what we mean, because it has enabled our languages to be the versatile tools that they needed to be. Now, descriptions can become more accurate by becoming more detailed, so long as they remain true. Can they also become more accurate by becoming truer?

If you said ‘that’s blue’ of a fading print, for example, would your words become truer as the print became bluer? Certainly, a description can become truer by becoming more detailed, as when we bring out the element of truth from some half-truth. So perhaps we should say that, in general, our words are true insofar as they describe the world. E.g. ‘snow is white’ is true insofar as snow is white. The famous biconditional – ‘snow is white’ is true if, and only if, snow is white – is either a special case of that, or else it presumes that the elements of truth and falsity can always be effectively isolated. (Clearly ‘snow is white’ is usually true enough, but snow can also be faintly blue, discoloured, transparent, or sparkling all the colours of the rainbow.) Now, the idea that there are degrees of truth is nothing new. That it’s only common sense is shown by such common phrases as ‘true enough’ and ‘very true’; and it has been formally explored by the so-called ‘fuzzy’ logicians [iii]. But I want to emphasise its prima facie plausibility here, because if statements can be, not just true or not, but also vaguely true – about as true as not – then the Liar paradox is easily resolved.

A simple Liar statement is ‘this is false’, which is false if true, but if false then not false (a statement is false when its negation is true). Since statements might be neither true nor false, a better example may be ‘this is not true’. And since sentences can mean different things in different contexts, an utterance of ‘what I’m now saying isn’t true’ is what we shall examine below. However, whereas modern introductions to the Liar paradox tend to be rather formal [iv], I shall be quite informal. Philosophers are in the business of clarifying things, and imprecision can of course lead us astray, so it’s unsurprising that many modern philosophers are fond of formal precision. But formal languages must get their meanings from natural ones. And an obscure formalism might also lead us astray. E.g., the use of biconditionals to define truth might go too easily unquestioned on a page of mathematical symbols; and for another example, see Curry’s paradox (below). So, let’s get back to informal alethic basics. (To be continued.)

[i] For a class of paradoxes that includes Grelling’s, see Thomas Bolander, ‘Self-Reference’, in Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy.

[ii] Alfred Tarski took that approach to formal languages (taking natural languages to be inconsistent). For details, see Wilfrid Hodges, ‘Tarski’s Truth Definitions’, in Zalta, op. cit.

[iii] Petr Hajek, ‘Fuzzy Logic’, in Zalta, op. cit.

[iv] J. C. Beall and Michael Glanzberg, ‘Liar Paradox’, in Zalta, op. cit.

Vaguely True Liars

2011-03-31T15:49:00.013+01:00

My next 4 posts explore, informally and briefly, the possibility that Liar utterances are about as true as not. Abstract:

I suggest that Liar statements – e.g. ‘this is false’ – are about as true as not. In other words, they are vaguely true, and vaguely false. And truth does seem to come in degrees (e.g. if a colour is about as blue as not, calling it ‘blue’ would be about as true as not). I also suggest that ‘this is not even vaguely true’ can be called ‘vaguely true’, even though it may then seem false, not just vaguely false. That’s because it is, more precisely, vaguely more vaguely true (similarly, a colour that’s roughly as blue-green as it is blue would usually be called ‘blue’ becaue it is a faintly greenish blue). I give similar resolutions to the paradoxes of Kurt Grelling and Keith Simmons, while I regard as of a different kind those of Bertrand Russell and Haskell Curry.

Links to the 4 posts:

......Introduction (via Grelling’s paradox)

......Liar statements are about as true as not

......My ‘revenge’ paradox (and Yablo’s paradox)

......Russell, Cantor, Curry, and Simmons’ paradox

An application of the above is a common sense refutation of the Divine Liar argument (against omniscience), see my Liars, Divine Liars, and Semantics revisited (in April’s issue of The Reasoner).

Curry's paradoxes

2011-03-18T15:14:00.005Z

The Liar paradox concerns utterances such as ‘what I’m saying isn’t true,’ which is, if true, not true, and which seems true if not. Another way of saying the same thing would seem to be to say ‘if what I’m saying is true, then pigs fly.’ Yet that utterance is paradoxical in a way so different that not only has it a different name – Curry’s paradox – it’s debatable whether its resolution should even resemble that of the Liar.
......Suppose I say ‘if what I’m saying is true, then P,’ where ‘P’ stands for any proposition. For simplicity, let’s say that C is the assertion that C implies P. Suppose, just suppose, that C is true. We are thereby supposing that C implies P. So we would also have P. That much is simple enough. Given C, and that if C then P, we get P. And yet that much is too much. If it’s true that by supposing C we also get P, then C really is true, and hence P is true, even though P could be asserting anything at all (even that pigs fly).
......That seems quite unlike the Liar. E.g. there was no ‘not’ in the previous paragraph: We didn’t consider C being either true or else not, and find both possibilities inadequate; nor did we see C seeming to say that it wasn’t something that C did indeed seem not to be. Rather, just by wondering what C was – in particular, whether C might be true – we seemed to get P. And so by deriving actual being from mere possibility, Curry’s paradox seems to be more like the Ontological Argument than the Liar.
......On the other hand, if it follows from the meaning of ‘not’ that either A or not-A, where ‘A’ stands for any proposition (e.g. that it’s raining), then it seems that if A implies B (e.g. that I’m carrying an umbrella), then either B or not-A (either I’m carrying an umbrella or it’s not raining). And conversely, if it’s the case that either not-A or B, then if it’s A (if it isn’t not-A), it must be B. So in short, C could seem to be asserting that either not-C or P. And then P effectively disappears if it’s false, leaving C asserting not-C.

Simmons' paradox

2011-03-16T11:25:00.009Z

The following, which is akin to the Liar paradox, is a paradox of self-reference by Keith Simmons, described on p. 231 of his ‘Reference and Paradox’ in JC Beall (ed.) Liars and Heaps (Oxford 2003):

Suppose I’ve just passed by a colleague’s office, and I see denoting phrases on the board there. That puts me in the mood to write denoting phrases of my own, and so I enter an adjacent room, and write on the board the following expressions:

......pi
......six
......the sum of the numbers denoted by expressions on the board in room 213.

Now I am in fact in room 213, though I believe that room 213 is my colleague's office. I set you the task of providing the denotations of these expressions.

Simmons goes through your (fictional) reasoning; here is what I think:
......The first expression, ‘pi’, referred to 3.14159..., and the second to 6. So if the third expression does denote a number, say N, then N = N + 9.14159... Given that by ‘number’ we mean finite number, it seems that the third expression can’t denote a number. So those three expressions denote only pi and six. But then the sum of the numbers denoted by those three expressions is 9.14159..., and so the third sentence does seem to denote a number after all. Or rather, it does because it doesn’t; and furthermore, it seems to denote 9.14159..., but therefore it seems to denote 18.283..., or rather, 27 and a bit, etc.
......I have, however, been implicitly assuming that reference is an all-or-nothing affair. Usually we can – and indeed, should – take it to be so, but is it so in general? Imagine, for example, a man staggering through a desert. He sees a mirage, which he takes to be a pool, and as it happens there is a pool, just where he takes one to be, but it’s obscured from his view by the mirage. As he staggers towards it, he’s constantly thinking ‘that pool looks cool’. As he nears the pool, its image gradually replaces the illusory one, without him noticing, so that the referent of ‘that pool’ gradually changes to the pool. And so at some point he may have referred only vaguely to it.
......Such a case would of course be exceptional, but so are scenarios designed to be paradoxical. And it does at least seem possible that the third expression of Simmons’ paradox referred only vaguely to 9.14159... It would also have referred, even more vaguely, to 18.283 (and so on), but while that’s even odder, it too seems possible. And if the alternative is paradoxical, vague reference may not be too odd. And such a resolution would cohere with ‘this is not true’ being vaguely true (about as true as not), and ‘is heterological’ being as heterological as not (see my previous posts this year).

God is Timeless

2011-03-06T15:27:00.007Z

When people say that God is timeless, they may mean many things. They may mean that He is above and beyond the mundane world, like the truths of mathematics, for example. But they would not then be disagreeing with Open Theism:
......Under Open Theism, God is certainly above and beyond His creation, a bit like a dreamer and the dream he finds himself within. And one of the few coherent philosophies of mathematics is Open Theistic Constructivism, in which God, being omnipotent, creates the truths of mathematics—from the basic concepts of a thing and of possibility (the latter grounded in His omnipotence)—doing so endlessly because such is the nature of the former concept, according to the resolution of Cantor’s paradox that takes it to be showing that cardinal numbers are collectively indefinitely extensible (one of the few coherent resolutions).
......Of course, they may instead mean that God is not changeable. But even that isn’t incompatible with Open Theism, under which God cannot change his essential properties. Nor can you, of course. You can’t become me, for example. You might change by becoming in a manner of speaking a different person (i.e. your character might change, for better or for worse), but under Open Theism God’s character remains perfect. So the only disagreement with Open Theism could be that such philosophers are denying that God could choose to cause any real change in anything; and not only is that clearly not what most believers mean when they say that God is timeless (is rather more like an absurd denial of the reality of change), such philosophers would be denying God’s omnipotence:
......The deliberate creation of anything contingent surely requires several real possibilities to choose between, as well as a single actuality amidst counterfactuals, and hence some sort of change (not necessarily one that takes place within spacetime).

Only the unfit evolve

2011-03-01T11:32:00.008Z

From Philosophy Now, I learn 'that people and organisations that miss their goals disastrously perform better in the long run.'

Professor Desai, who led the study, said “knowledge gained from success was often fleeting while knowledge from failure often stuck around for years.” [...] He says failure causes a company to search for solutions and it puts the executives in a more open mindset. He doesn’t recommend seeking out failure in order to learn.

...which makes me think of the merchant bankers. Will they perform better now? Only if we make them (despite our politics being dominated by the short-term), I think. You know, their performance was always chaotic, even in the good old days, as the new maths of chaos showed us in the 80's. But they took past success to be indicate a propensity for success; ironically, they were bad at applied math.
......I find that ironic because it was pointed out to me, when I was reading physics (in the 80's), that merchant banking was the sort of 'good job' that a physics degree qualified one for. (That was what made physics such a good degree.) At the time, I wondered whether we really would've been a better society had more women wanted to play such games. (Margaret Thatcher, with her Chemistry degree, hardly thought of merchant banking as a waste of a physics degree.)
......What can you do? The young are always with us. And we can hardly want to dilute democracy, but why not give more experienced voters additional votes? Perhaps we shouldn't take the vote away from those who've shown themselves to be very selfish (unless we've locked them up and thrown away the key), but why not give an extra vote to all those who haven't (yet) shown themselves up; indeed, why not give extra votes to those who deserve honours? Wouldn't that be fairer? (It would certainly be safer.)

Philosophers' Carnival #121

2011-02-21T00:01:00.021Z

...you're a philosopher. So welcome to Philosophers' Carnival #121. Not so much a one-to-one as a party (a rather Platonic party). The bouncer's been busy (arguably too busy, or not busy enough). And there's some of the hard stuff later on; so, enter...

......the Hallway (epistemology)

Keith DeRose imagines 'Nico at the Zoo with Zebras' (at Certain Doubts), an example of how knowledge attributions can sometimes be true even when they’re about insensitive beliefs.

John Wilkins compares Elliott Sober's 'Modus Darwin and the *real* modus darvinii' of 'affinity, explained by common ancestry' (at Evolving Thoughts), showing that the former should've been the latter.

Maryann Spikes thinks of 'Atheism and agnosticism (really, apisticism) as belief' (at Ichthus77), and also thinks that you can only be apistic if you don't claim to be.

......the Games-room (logic and language)

Ben Nelson wonders about 'Trust as a truth-maker' (at Talking Philosophy). Of course, "X trusts me" is made true by X trusting me, but Nelson takes a broader (even deeper) look at this kind of thing.

I ask 'Is 'pretty' pretty?' (It is, and it isn't:)

Matt asks 'How slippery is the slope?' (at The Consternation of Philosophy), and concludes that 'the slippery slope fallacy is a slippery beast, and is perhaps best not thought of as a fallacy at all.'

......the Dining-room (metaphysics)

Edward Feser asks 'Why are (some) physicists so bad at philosophy?' (at Edward Feser), and...

Eric Steinhart wonders 'Why Materialism is Unscientific' (at Camels with Hammers), both in response to astrophysicist Ethan Siegel asking 'Can You Get Something For Nothing?' (both answer No).

Jeremy Stangroom wonders, 'A First Unmoved Mover?' (at Talking Philosophy). He shows Copleston and Russell describing the Atomists differently; not because either was bad at the history of philosophy, but because good answers to the question Why should there be something rather than nothing? do include Because the metaphysically necessary being is perfectly good as well as the (now) obvious No reason.

......the Living-room (mind)

Kenny Pearce knows that 'Sometimes it's Rational to act Arbitrarily' (at Kenny Pearce). 'In ordinary cases it is irrational to take a certain course of action when you know there is a better one available to you,' but what if you are asked to choose any natural number of dollars? (Sobel thinks that choosing anything would still be irrational; why would he think that?)

Constantine Sandis entertains 'Enchanting Causes' (at Flickers of Freedom), and so 'tests our intuitions about what sort of desire makes an action intentional.'

Joel considers 'Killing a Vegan: Degrees of Subjectivity' (at Florida Student Philosophy Blog), arguing that chickens (as opposed to Vegans) may not feel phenomenological pain, because they don’t have the 'I' concept, or the neurological ability to do much more than react physically.

......the Kitchen (moral philosophy)

Robin Hanson asks 'What Virtue Privacy?' (at Overcoming Bias), and by discussing Thomas Nagel's 'Concealment and Exposture' argues 'that humans had huge heads to subtly evade social norms while pretending to enforce them.'

Tim Dean considers 'Morality, Health and Sam Harris' (at Ockham's Beard), arguing that Harris's moral realism makes naturalism harder to defend, and suggesting that we could just say that 'Being animals, we pursue health. And being social, we pursue morality.'

James Gray defends moral realism against Hume, by considering 'Intrinsic Values & Beliefs About Reality' (at Ethical Realism).

Antti Kauppinen explains 'How the Experience Machine Works' (at Experimental Philosophy), before objecting to Felipe de Brigard’s recent ex-phi objection to Robert Nozick’s result.

Richard looks at 'Natural Agents and Status-Quo Bias' (at Philosophy, et cetera), questioning Carolina Sartorio, who argued (via Trolleys) 'that we need stronger reasons to justify interfering in a process (e.g. deflecting a trolley) than to justify abstaining from such involvement.'

Clayton Littlejohn has 'Ethical Intuitions (II): Cosmic Coincidence' (at Think Tonk), the second in a series of posts on moral epistemology (some empirical arguments having been considered in 'Ethical Intuitions (Part I)'): A version of intuitionism on which moral properties supervene upon natural properties is defended against Matthew Bedke.

Jussi Suikkanen conjoins 'Deliberative Contractualism and the Conditional Fallacy' (at PEA Soup), arguing that the former (by Nicholas Southwood) commits the latter.

Thom Brooks announces 'Thom Brooks on "Punishment: Political, Not Moral"' (at The Brooks Blog). British Hegelians make Alan Brudner's retributivism more attractive, apparently.

Chris Bateman considers 'A Categorical Imperative for the Other' (at Only a Game), suggesting that 3 formulations of Immanuel Kant's categorical imperative are more easily seen to be equivalent if they (or something like them) are derived from Emmanuel Levinas' concept of the Other.

Anders Sandberg wonders how much 'Intolerance we ought to encourage?' (at Practical Ethics). 'At the very least we can make it a social rule that just as we frown at racist, sexist or homophobic statements we frown at pseudoscience or deceptive evidence.'

......the Backyard (other)

Paul Newall examines different views of 'Astrology and its problems: Popper, Kuhn and Feyerabend' (at The Kindly Ones), and suggests that 'the philosophical problem for astrology is not that it can always explain failures (Popper) or that it does not attempt to solve problems (Kuhn) but instead that it has stagnated (Feyerabend).'

Kieran Healy looks at 'Gender divides in Philosophy and other disciplines' (at Crooked Timber). More than 70% of US PhDs in psychology were awarded to women, and it's 60% in sociology. Still, it's only 40% in political science, and 30% in philosophy.

Brian Leiter also looks at 'Women in Philosophy in the US' (at Leiter Reports), and finds that the proportion teaching philosophy is only about 20%. That's pretty good, given our context (it's more than 5 times the proportion of female bloggers here).

Gary Williams has some 'Thoughts on Cordelia Fine's new book Delusions of Gender' (at Minds and Brains); e.g. 'Maybe 1000 years in the future there will be an equal amount of male and female physicists, philosophers, and computer scientists,' because our brains are (equally) plastic.

James Warren reveals 'Rejection letters of the ancient philosophers' (at Kenodoxia). Bitchin'

.........is there 'More on philosophy and society'? No, because the party's over (hopefully before the fighting starts). Almost all-male, and the kitchen the most popular place; what a party. But if this carnival bored (or annoyed) you, or if your entry bounced (for no good reason), the solution is to host a carnival, which you should also do if you liked this one, of course: No hosts = no carnivals.
......And whenever you find yourself reading an interesting post, of a philosophical nature, you should submit it, because no posts = no carnivals. Carnival #122 will be at Ichthus77

Is ‘pretty’ pretty?

2011-02-19T23:19:00.017Z

Are any words pretty? Maybe not outside of calligraphy (or song), but on the other hand, ‘pretty’ isn’t too odd-looking, as words go. And it does make us think of prettiness. So I’m reluctant to say that it isn’t pretty, i.e. that it’s heterological (that it doesn’t describe itself accurately). Is it, then, that it isn’t heterological (that it describes itself accurately), that it’s a pretty predicate? I wouldn’t go that far; to me, it seems only vaguely pretty. And so it seems to me that ‘pretty’ is vaguely heterological.
......Is being heterological a matter of degree? Well, descriptive accuracy does seem to be. E.g. “is short” is a fairly short predicate expression, while “is so far from being extremely long that, not only is it not very long, it’s short” clearly isn’t. So there’s probably an expression that means the same as “is short” and which is vaguely short (unless an expression N letters long can be short while one N + 1 letters long isn’t), and hence vaguely heterological. Furthermore, is ‘boring’ boring? Not very boring now I come to think about it. Etc.
......So, since it’s a matter of degree, should we not say that predicates are heterological insofar as they don’t describe themselves very well? If so then ‘heterological’ is as heterological as not, is (only) vaguely heterological. And Grelling’s paradox—that ‘heterological’ is hetrological if, and only if, it isn’t—does rule out the non-vague extremes. I’m not suggesting that ‘heterological’ is neither heterological nor not—i.e. that it’s not heterological and that it is—however, because if I’m right, such trivalent (or dialethic) claims are inaccurate, aren’t very true (nor very false).

Classical Logic, how is it correct?

2011-02-12T18:05:00.009Z

According to Stewart Shapiro, Classical Logic is first-order (and hence formal) predicate logic, which he describes in some detail (in that SEP entry), having first noted that:

Formal languages, deductive systems, and model-theoretic semantics are mathematical objects and, as such, the logician is interested in their mathematical properties and relations. Soundness, completeness, and most of the other results reported below [in that SEP entry] are typical examples. Philosophically, logic is the study of correct reasoning. Reasoning is an epistemic, mental activity. This raises questions concerning the philosophical relevance of the mathematical aspects of logic. How do deducibility and validity, as properties of formal languages--sets of strings on a fixed alphabet--relate to correct reasoning? What do the mathematical results reported below [in that SEP entry] have to do with the original philosophical issue?

Shapiro goes on to list some possibilities; e.g. perhaps "the components of a logic provide the underlying deep structure of correct reasoning."
......Another possibility is that "because natural languages are vague and ambiguous, they should be replaced by formal languages," or rather (since formal languages are all defined using natural languages) "regimented, cleaned up for serious scientific and metaphysical work." Now, scientists often do define their own scientific terms; but how could that process apply to logic? Informal logic must be good enough for us to work out the correct formal language to use, if there is one (otherwise our justifications would become circular). "Another view is that a formal language is a mathematical model of a natural language in roughly the same sense as, say, a collection of point masses is a model of a system of physical objects." Such a view makes sense, in view of the many logics studied by logicians; but what, then, can we say about logic in the sense of correct reasoning? We use such a logic when we use any mathematical model scientifically; we must reason correctly about the model. Is classical logic a good model of informal logic?
......Despite the various logics that logicians work on, most arguments are presented in a classical logical style (even those about other logics). So presumably we do think that such components cut our reasoning pretty close to its joints, so to speak. Is classical logic correct? Some philosophers say so (e.g. Alexander Pruss recently gave that as a reason for rejecting open future views, in comments on this post of his), but I wonder how it is. It's hard for me to specify my worries without having already resolved them; but let's look at some simple logical arguments, such as might be used to introduce classical logic, and see how they may fail to be examples of classical logic (without going too far into the related problems of metaphor and vagueness).
......Grass is green, and all flesh is grass, so, is all flesh green? That's clearly invalid, the second premise being metaphorical; but did we use classical logic to work that out? And suppose I was holding something that wasn't green; could I deduce that it wasn't grass? Again no, because it's not true that all grass is always green. But what if all Blurps were always green; could I deduce that I wasn't holding a Blurp? I don't see why not. And surely I could put that argument in classical logical form. And yet if "is green" is a classical predicate, then not only what I'm holding, but anything and everything is either green or else not green (LEM); whereas there's clearly a shading from green, through greenish and vaguely greenish colours, to those that aren't green. Is it that classical logic is a good model, but that informal logic (real logic) must be slightly different? If so, what's the point of all that maths that Shapiro introduces (rather well)? I'm not saying there's no point to it. (I'm hoping it's not turning philosophy into a pseudo-science.) I'm rather asking whoever's reading this post, what do you think about classical logic?

Reasoning badly from Yablo’s paradox

2011-02-07T08:02:00.009Z

Paradoxes can be hard to resolve, so it can be hard to reason well from them. A nice example is a recent argument that the past is finite, by Laureano Luna (2011: ‘Reasoning from paradox’, The Reasoner 5(2), 22–23). I shall vary the details. Suppose there’s a place where, once a year, every year, someone says “No previous utterance here was true,” with nothing else ever having been said there. Details can be varied, so long as we would, were the past infinite, have an infinite sequence of similar utterances. Indeed, it’s because we can vary the details that the following contradiction seems to follow from supposing the past to be infinite (rather than from, say, supposing that language need not begin with evident truths).
......Each utterance in that place concerned the past, so it seems that each utterance should be either true—were none of the previous utterances there true—or else false—were at least one of them true. But none of them can be either, on pain of Yablo’s paradox: Were no utterance there true then, via what each said, each would be true (and not true); but were any of them true, then since none of the earlier ones would have been true, its immediate predecessor would also, via what it said, have been true (and not true).
......But even if such sequences of utterances are impossible, the past might be infinite. One possibility is that simply infinite sequences, e.g. the natural numbers, are indefinitely extensible (are Potential Infinite) in the sense that while there’s always a next element, e.g. a bigger number, there’s no complete collection of them all. Standard mathematics assumes that such isn’t the case, but we’ve yet to discover that it isn’t. And if it is, then although we naturally think of past years as stretching back in time forever, the past couldn’t be the whole of such an infinite sequence, and so our infinite sequence of utterances would’ve been impossible too. And yet the past might, even so, be infinite. E.g. there might have been, before the Big Bang, some infinitely slow process, which took an infinite time to complete (and before which there might have been something else, possibly with no beginning); such a process has an infinite duration in the sense that we might go any natural number of years back into it and not reach its beginning, and also in the sense that were it the unit of time, all the time since the Big Bang would be relatively infinitesimal.
......Another possibility is that truth, or descriptive accuracy, is essentially a matter of degree. We might take an utterance of “No previous utterance here was true” to be asserting that none of those previous utterances described the past well enough for it to be classed as true. Yablo’s paradox would then be ruling out every possibility except the possibility that all those utterances described the past only vaguely, that they were all vaguely true. (Does it seem that they would then have been failing to describe the past well enough to be classed as true? If so, note that the suggestion is that either classification—true or not—would be less accurate than that of vaguely true.) So, our contradiction may well have been due to our having used, in effect, a rather artificial language. So we seem to have shown only that either the language of those utterances doesn’t allow such sequences of sentences, or something else (e.g. maybe the past must be finite, or maybe the natural numbers are indefinitely extensible).

Philosophers' Carnivals: Now & Next

2011-01-31T01:00:00.011Z

Philosophers' Carnivals "showcase the best philosophical posts from a wide range of weblogs," as it says on the carnival's homepage. From today, carnival #120 is at nicomachus.net. And carnival #121 will be here in 3 weeks time, so if you find yourself reading something nicely philosophical, posted between now and then, please consider submitting it, via the online submission form, even if you wrote it yourself: "Don't be shy, we want to hear from you, that's the whole point of this project! Your post doesn't need to be anything earth-shattering - it just needs to be something that other philosophically-minded people might enjoy reading."
......As for what you can submit, there are No Rules, except: "No self-help, mysticism, marketing spam, etc." Of course, marketing spammers are unlikely to have bothered reading as far as this, so telling them not to bother submitting seems pointless. And I wouldn't rule out what some academic philosophers like to call 'mysticism', e.g. Mathematical Platonism, Substance Dualism, Open Theism and so forth (since such is just realistic metaphysics). Nor shall I reject whatever formalized craziness such academics work on instead, of course (since I should be unbiased in my hosting). Indeed, since the number of the carnival will be 121 (which sounds like "one-to-one") there's even some hope for self-helpers (and Continental Philosophers) whose positive thinking has carried them thus far, because insofar as their posts describe how the ideal of the Socratic Dialogue relates to their brand of self-help (or Derrida) I shall look upon them kindly.
......Here's a cautionary tale about rule-following: Many years ago, a port on the east coast was industrializing. To its north and south were two large estates, the country houses of two progressive squires, who built factories and docks in the port, and cheap housing for their workers there. Peasants to the west of the port flocked there, to earn more and to be free from their old-fashioned and relatively oppressive squire. As his peasants deserted his lands, that squire soon found himself with cashflow problems, and eventually he was reduced to opening his mansion to the public. He even built an inhumane zoo in its overgrown grounds; but things got no better. He got more and more depressed. One day he became quite deranged, and smashed up his zoo. Then he climbed onto the back of a huge hippopotamus and rode it towards the port. Now, the two rich squires heard of him crashing through their workers' slums, but they were unable to stop him because he had the law on his side, the law which states that the squire on the hippopotamus is evil to the slums of the squires on the other two sides.

Liars, Divine Liars, and Semantics revisited

2011-01-25T12:14:00.018Z

Divine Liar arguments aim to show that there’s no omniscient being—that no one knows all that’s true—in the following way. Suppose I say “No omniscient being knows that what I’m now saying is true.” If (as I believe) no one is omniscient, then no omniscient being exists, to know anything. So in that case, what I said was true. What I said was therefore an assertion, whether it was true or not. And if it wasn’t true—if it’s not the case that no omniscient being knows that what I said was true—then some omniscient being knows that what I said was true, despite it not being true, which is impossible (knowledge being of truths). So I asserted a truth; and so either that was a truth that some omniscient being doesn’t know, which is also impossible, or else there’s no such being. ......However, resolutions of the Liar Paradox might show that such arguments are invalid, e.g. according to Daniel J. Hill (2007: ‘The Divine Liar Resurfaces’, The Reasoner 1(5), 11–12) and my earlier article (2008: ‘Liars, Divine Liars and Semantics’, The Reasoner 2(12), 4–5). So, suppose I say “What I’m now saying isn’t true.” If what I said was true then, as I said, what I said wasn’t true. Does it follow that what I said wasn’t true? The paradox is that if so, then since that’s what I seem to have said, I seem to have said something true. The resolution defended earlier by me (2008) takes my utterance to have been meaningless, so that I didn’t really say anything. But we may then wonder how it was that it seemed so clear what my utterance would have meant had it been true; and my Divine Liar utterance was even more obviously meaningful. Another popular resolution would regard my Liar utterance as equivocal, with the word ‘true’ naming many different predicates in Hill’s (2007) Tarskian hierarchy. But formal languages can only be defined via natural language; and my informal Divine Liar utterance wasn’t obviously that equivocal. ......Questions of truth are essentially questions of how well our words are describing the world. So insofar as my Liar utterance wasn’t meaningless, it was asserting that it wasn’t describing itself very well, not well enough for it to have been true. And since it was nothing if not self-contradictory, it certainly wasn’t describing itself very well. But therefore, in view of what it was asserting, it seems to have been describing itself quite well after all. Was it describing itself well enough for it to count as true? I’m reluctant to call it ‘true’ as follows. If it was true because it wasn’t, then it was true and not true, but surely something’s only not some way if it’s not the case that it is. Nor do I want to say that it was neither true nor not true, as that’s just to say that it was not true and also true. Nevertheless, my utterance wasn’t describing itself very well, and was therefore describing itself quite well; so perhaps it was only partially true. If so then calling it either ‘true’ or ‘not true’ would both be inaccurate, would both be only partially true. ......We naturally focus upon whatever truth we can find in what people say, or upon an obvious untruth. And things are usually described accurately enough for some obvious purpose, or not accurately enough. But would it be unrealistic to think of truth (descriptive accuracy) as a matter of degree? The classic example is that of Vann McGee (1991: Truth, Vagueness, and Paradox, Hackett, 217): If “Harry is bald” is true insofar as Harry is bald, ‘true’ should be at least as vague as ‘bald’. And quite generally, why should we believe that our words are much better defined than our purposes have required them to be? Maybe natural language has a ubiquitous—since usually unobtrusive—vagueness. (That would explain why the discovery of a contradiction so naturally triggers an attempt to clarify our terminology.) And in particular, the Liar Paradox might be revealing this ordinarily obscure vagueness of ‘true’. That’s because if my Liar utterance was only partially true, then it would follow from what I said only that it was also partially not true, which clearly coheres with it being only partially true. There’s no inconsistency—no more paradox—and it seems that much the same could be said of any Liar sentences. ......And if that is how the Liar Paradox should be resolved, then my Divine Liar utterance would have been only partially true if there is an omniscient being. My Divine Liar argument was therefore fallacious, because arguments should have premises that are unequivocally true enough to count as true under all relevant hypotheses. But if you asked an omniscient being whether my Divine Liar utterance was true, she might say that it contained an element of truth. That might be a more informative—more true and less misleading—answer than a simple ‘yes’ or ‘no’. ......Similarly, the best answer to the question “Is this colour blue or not?” could be to say that it’s vaguely bluish. Ordinary objects are almost always either blue or not, but colours don’t really divide into those that are blue and those that aren’t. On the two sides of any such line, between the blue and the other colours of some spectrum, would be colours that were indistinguishable. So there’s no such division; and so there’s some colour of which, rather than saying that it’s blue, or that it isn’t, we ought to say that it’s bluish. Note that such a colour might look blue against a background of colours that weren’t blue, or even if you just wondered whether it belonged to that class of colours, and so postulated it amongst them (cf. what we find paradoxical about the Liar Paradox). ......Incidentally, some formal work on ‘true’ as a vague predicate is well described as Fuzzy Logic.

Liars Are Fairly True

2011-01-01T11:44:00.004Z

Suppose I say “what I’m saying isn’t true.” If what I said was true, then as I said, what I said wasn’t true. Does it follow that my words weren’t true? The famous paradox is that if so, then since that’s what I seem to have said, I seem to have said something true. A fairly popular resolution takes my words to have been meaningless, so that I didn’t say anything. But if my words had been meaningless, you could hardly have known what they would have meant had they been true. Is our ordinary conception of truth shown by such Liar-style sentences to be deficient? Let’s see why not.
......To begin with, such sentences are in some ways like Truth-teller-style sentences. If I said “what I’m saying is true,” for example, what would I be saying? Not much. Questions of truth are essentially questions of how well our words describe the world, and “this is a good description” isn’t much of a description. Still, it might not be too bad a self-description, precisely because there isn’t much to describe. If someone saying “what I’m saying is true” intended to be speaking the truth, should we deny that she was telling the truth? It may be hard to say, but therefore it might be that such sentences are not so much vacuous as vague. Since “what I’m saying isn’t true” also addresses nothing but its own descriptive power, might it also be, in its own way, rather vague? Consider the following analogy.
......If I said of some colour, “I wouldn’t say that it’s blue,” I might not be saying that it wasn’t blue, because colours don’t divide into those that are blue and those that aren’t. To see that, consider a spectrum: On the two sides of any such line, between the blue and the other colours, there would be colours that were indistinguishable. So there’s no such division; so there’s some colour of which, rather than saying it was blue, or that it wasn’t, I’d prefer to say, more precisely, that it was bluish but not very blue. (Since the perception of colour is subjective, you might say it was blue, or that it wasn’t.) Our perception of colour is also context-sensitive, e.g. it’s affected by surrounding colours, and by our preconceptions. So if I wondered if our colour really was blue, I might thereby see it as not blue, while if I then wondered if it was therefore not blue, it might seem pretty blue (even to me).
......And similarly, it’s when “what I’m saying isn’t true” has been thought of as definitely not true that it seems most clearly to be true. More precisely, while those words aren’t giving us a very good description of their own meaning—they’re self-contradictory—we therefore have a description that isn’t too bad, insofar as it’s saying that it’s not a very good description. In short, they’re rather nonsensical (and false), but therefore fairly true (and false). And that’s basically how Liar-style sentences are compatible with our ordinary conception of truth. We need a bit more clarification, but it should soon become clear that while we can always be more precise, there’s no threat to truth here.
......What is truth, if not a sufficiently accurate description? Usually we describe things accurately enough for some obvious purpose, or else we don’t, so we tend to assume that truth is black-or-white. But it’s really a matter of degree, in a context-sensitive way. E.g. the table at which I’m writing this is flat enough for that purpose, so “this table is flat” is true enough, but might be false were I writing about geometry. And in general, our words tend not to be much better defined than our purposes have required them to be. So natural language has a ubiquitous—since ordinarily unobtrusive—vagueness (whence the way to resolve paradoxes, and uncover other fallacies, usually involves clarifying some terms). Of course, the words of “what I’m saying isn’t true” have clear enough meanings, so there’s no simple equivocation to discover. But it should help us to resolve the paradox if we don’t demand anything too unrealistic. (Similarly, we shouldn’t demand that colours be either blue or else not blue.)
......Liar-style sentences present themselves as misrepresenting themselves, so their meaning is self-undermining. And they can be read (or heard) in two basic ways—each a necessary part of the other’s context—because their meaning self-undermines in a loopy sort of way. Insofar as Liar-style sentences are true they’re also false, and they need concern nothing but their own truth, so they can certainly be read as nonsensical. But they’re not just senseless, and hence not at all true, because insofar as they’re not true they’re easily read as true. So they also have that sense. But they can’t be nothing but partly true and hence partly false, because that would leave nothing for them to be true or false about.
......This resolution—that Liar-style sentences are fairly true, in that loopy way (they’re fairly true because they’re rather nonsensical, and they’re rather nonsensical because insofar as they’re true they’re also false)—is a strengthened version of the resolution that takes them to be nonsensical. So for those who believe that an omniscient being is logically possible, it allows a similar reply to Divine-Liar-style sentences. E.g. the problem with “no omniscient being knows this” is that it can’t be true if there’s an omniscient being, but if it isn’t true then, since no one could then know it, it would seem to be true. My new reply is that if it’s only fairly true (in this loopy way) then no epistemically perfect being would have to know it, except to know it for what it is. And note that “no omniscient being knows any of this” is simply false, e.g. such a being would know those words. (Similarly, “what I’m saying isn’t at all true” is fairly false.)

Omniscience Again cont.

2010-12-17T08:13:00.008Z

This is the last of 17 posts, which are collectively Eternity, etc.
......There is something counter-intuitive about the suggestion of the previous post, of course (even on the modern view of arithmetic). If B is the biggest Beth that has been constructed, then my suggestion denies that 2-to-the-power-of-B exists, where 2-to-the-power-of-B is the cardinality of P(X) when X has cardinality B. Were there no such B, my suggestion would deny that the union of the existing sets has an actual cardinality, on the modern view of the natural numbers (on the older view, it would deny the existence of M + 1, where M is the biggest natural number divinely constructed). Either way, my suggestion is effectively that there are true statements that God did not know but which were bound to be true and which, if we could come to know them, we would most naturally say had always been true. Intuitively, that seems to fall short of divine omniscience.
......Nevertheless, we know from section III that what can seem, with hindsight, to have been timeless truths may not have been. And according to section VII it is logically, not just physiologically, impossible for anyone to say or know all such things. Such statements therefore belong to an indefinitely extensible totality. Would it therefore be more accurate to talk of possible statements here? Maybe not [i], but it’s certainly logically possible that our intuited shortfall is due to our being dependent creatures. For us, even physics is immutable, but God is certainly the ground of metaphysical possibility. And He may well be the ground of all meaning and value. So the counter-intuitiveness of divinely created mathematics may prove, upon reflection, to be no more conclusive than the counter-intuitiveness of Divine Command Metaethics [ii]. After all, my suggestion does not deny that, in the time we took to think of 2-to-the-power-of-B, God had already constructed it [iii].
......So, to recap, God’s omnipotence conflicts with His timelessness, according to section VII, unless we deny arithmetical Realism, or deviate further than Presentism does from standard logic. And under Presentism, even such an omnipotent God could be necessarily omniscient. So what follows from God being necessarily omniscient—and our freedom being libertarian—is primarily disjunctive. Either God has timeless knowledge of the future—if that does cohere with our freedom being libertarian—but Realist arithmetic is paradoxical, or Realist arithmetic is divinely constructed and time is Presentist, or some other option. So even if they regard God as necessarily omniscient, Perfect Being Theists who take a libertarian view of free will—and regard the future as (partially) real—should not reject Open Theism.
......Notes:
......[i] Statements are basically possible assertions (see note ii of Eternity), but a possible statement is not necessarily just a statement. Similarly, one might be unable to say something in French, and yet be able to learn (more) French, so that one would be able to be able to say it. It is of course hard to tell how apposite that analogy is, for the language (so to speak) of God’s thoughts.
......[ii] Lois Malcolm, “Divine Commands,” in Gilbert Meilaender & William Werpehowski (eds.), The Oxford Handbook of Theological Ethics (Oxford Univ. Press, 2005), pp. 112–29.
......[iii] Suppose (see note v of Possible Worlds) that a God who could change had made our 4-dimensional world in an instant. Then some biggest Beth, say B, would be known by Him at all (of our) times. But then we might use “2-to-the-power-of-B” as a definite description of a Beth that He does not know, at any (such) time, which hardly coheres with His being the greatest conceivable being. By contrast, a Presentist God would most plausibly be learning arithmetic too quickly for us to describe any such number.

Omniscience Again

2010-12-15T08:08:00.004Z

This is the sixteenth of 17 posts, which are collectively Eternity, etc.
......You may be wondering how, if the Open God is forever acquiring arithmetical knowledge (see previous post), He could ever be omniscient (or how His understanding of His options could ever be perfect). It would not even help us here to think of omniscience in Swinburne’s terms, because however much arithmetic God knows it is logically—indeed, metaphysically—possible for Him to know more (according to section VII); and nor could He know all the interesting Beths (that could ever exist) [i], because the smallest Beth that He did not know would be of some objective interest.
......Nevertheless, such Perfect Being Theists as Augustine and Duns Scotus took the Platonic Forms to be divinely created (in view of God’s omnipotence) [ii], and similarly, Open Theists might take arithmetic to be divinely constructed [iii]. Suppose that arithmetical statements are true or false only when divinely proved or refuted (respectively). That process could not always be instantaneous, according to section VII, and of course, not yet knowing something that is not yet true would not obstruct omniscience. And while most of us think of arithmetic as timeless, by “arithmetic” we ordinarily mean finite arithmetic, and on the modern view such a God could have always known all of that (instantaneously constructed in His primal state). Indeed, He could have always known all the Beths that are not, for us, unimaginably large.
......My suggestion is therefore that God, being epistemically omnipotent, constructs each and every modal consequence of the concept of a thing (which He understands perfectly), and in particular the cardinalities of possible creations (doing so endlessly because such is the nature of that concept). He knows all the Beths that exist. Before He constructs a Beth, it has only a potential existence. And eventually (and arbitrarily quickly) He knows any Beth that could ever exist. And His understanding of such possible Beths is perfect (cf. how we could understand the essence of an arbitrary natural number, even on the older view of arithmetic).
......Notes:
......[i] For a similar suggestion, see Menzel, “God and Mathematical Objects,” pp. 93–4 n. 42.
......[ii] For Augustine, see Sorabji, Time, Creation and the Continuum, p. 252. For Duns Scotus, see Gunton, The Triune Creator, pp. 118–9. For more details, see Copan & Craig, Creation out of Nothing, pp. 173–80.
......[iii] Menzel, “God and Mathematical Objects,” defends such a view, called “theistic constructivism” by Copan & Craig, Creation out of Nothing, p. 191.

Cantor’s Paradox again

2010-12-13T08:00:00.005Z

This is the fifteenth of 17 posts, which are collectively Eternity, etc.
......Because of all those unions (see previous post), our collection is a nested hierarchy of sets, whose cardinalities the Beths are defined to be. And so because our collection is not quasi-spatial, nor are the Beths, collectively. So even on the modern view, cardinal arithmetic is indefinitely extensible [i]. And while that result is of a kind with Cantor’s Paradox [ii], it is the belief that cardinal arithmetic is timeless that makes it paradoxical. The atemporalist faces some tough choices, because the truths known by a timeless God would be collectively quasi-spatial, rather than variable.
......But an everlasting God could acquire arithmetical knowledge endlessly. And Presentist time is merely our natural reification of the possibility of change, not a real dimension. And under Presentist Open Theism, that possibility originates with the greatest conceivable being’s power to change. So such a God would have enough time to know each arithmetical truth, and to know it arbitrarily quickly. Time would then be indefinitely extensible, and absolutely continuous [iii], in the sense that for any duration, and for any Actual Infinite cardinal number (that could ever exist), that duration has more than that many instants (possible instantaneous changes).
......It seems, then, that only a God with the power to change is, for every Actual Infinite cardinal number, able to know all about a possible world of so many things, and hence able to create such a world perfectly freely (with a perfect understanding of His options). So the argument at the end of section VI becomes an argument that God is, since omnipotent, not timeless. And note that the informality of this rather mathematical section does not make that a weak argument. Formal proofs can only prove theorems within axiomatic systems, and since the justification of such axioms is necessarily informal, so informality also suits a more direct argument about metaphysically possible creations.
......Notes:
......[i] For more details, see W. D. Hart, “The Potential Infinite,” Proceedings of the Aristotelian Society 76 (1976): 247–64; Alvin Plantinga & Patrick Grim, “Truth, Omniscience, and Cantorian Arguments: An exchange,” Philosophical Studies 71 (1993): 267–306; Stewart Shapiro & Crispin Wright, “All Things Indefinitely Extensible,” in Agustin Rayo & Gabriel Uzquiano (eds.), Absolute Generality (Clarendon Press, 2006), pp. 255–304; Nicholas Rescher & Patrick Grim, “Plenum Theory,” Noûs 42 (2008): 422–39.
......[ii] For Georg Cantor, sets were consistent Actual Infinite collections. But he thought that all Potential Infinite collections presuppose Actual Infinite collections, much as mathematical variables range over fixed domains. So he thought of collections like that of all the sets (Cantor’s Paradox) or all the cardinal numbers as Actual Infinite but inconsistent. For more details, see Michael Hallett, Cantorian set theory and limitation of size (Clarendon Press, 1984), pp. 24–48. Of course, taking inconsistency on the chin like that is a high a price to pay for Realism (whence the foundation of mainstream mathematics is now an axiomatic set theory). But even if Potential Infinite collections do depend upon something being Actual Infinite, that might be a power (see note iv in Cantor's Paradox) or a length (see following note) rather than a collection.
......[iii] For such continua, see my “To Continue with Continuity,” Metaphysica 6 (2005): 91–109; Philip Ehrlich, “The Absolute Arithmetic Continuum and its Peircean Counterpart,” in Matthew Moore (ed.), New Essays on Peirce’s Mathematical Philosophy (Open Court, forthcoming).

Cantor’s Paradox cont.

2010-12-11T07:53:00.006Z

This is the fourteenth of 17 posts, which are collectively Eternity, etc.
......You may be familiar with N (see previous post) from school mathematics. Such informal sets are basically collections that are quasi-spatial, in the sense that their members coexist (insofar as they do exist) altogether. Given any spatial collection—e.g. some ordinary objects in a room—any sub-collection of them is clearly also spatial; and similarly, a definitive property of our informal sets is that every conceivable sub-collection of such a set is itself quasi-spatial, is a subset [i].
......Surprisingly, the modern view (of arithmetic as timeless) offers little support to atemporalism, as follows. To say that two collections have the same cardinality—the same cardinal number of members—is to say that the members of each collection could all be paired off, one-to-one, with those of the other [ii]. And for any set, S, if the collection of all its subsets is also quasi-spatial, then that collection—including (for simplicity) the so-called improper subsets, S and the empty set—is the powerset of S, say P(S). And according to Cantor’s Diagonal Argument [iii], P(S) always has a greater cardinality than S.
......In particular, the cardinality of P(N)—which Peirce called “Beth-1”—is greater than the cardinality of N, which is Beth-0. And the cardinality of P(P(N)) = P-squared(N) is Beth-2, which is greater than Beth-1. And so on; for each natural number n, P-to-the-nth-power(N) has Beth-n members. And the union of N and all those P-to-the-nth-power(N) is the collection of all their members. For each n it contains at least Beth-(n + 1) members. So its cardinality, say Beth-omega [iv], is greater than Beth-n for every n. And if that union is also a set, say U, then by Cantor’s Diagonal Argument, P(U) has an even greater cardinality, Beth-(omega + 1) [v],
......We might expect that union to be a set, because Beth-0 being an Actual Infinite number means that all those Beth-0 sets coexist quasi-spatially (like a row of houses, whose contents therefore coexist similarly). So the next union might be of U and all the P-to-the-nth-power(U). But by continuing in that way, taking powersets and unions as far as is logically possible [vi], we cannot end up with a set because from any set we could have continued further in that way. We have, then, a collection that is not quasi-spatial, being generated by a process that cannot be completed (as a matter of logical necessity). Continued...
......Notes:
......[i] By contrast, if the natural numbers are forever growing, according to the rule of add 1 repeatedly, then only those sub-collections that are similarly specified by a finite rule exist in the same kind of way.
......[ii] The natural numbers are finite cardinal numbers. And N has the same infinite cardinality as the subset of just the even numbers because n can be paired with 2n for all natural numbers n. There are, in an obvious sense, more natural numbers than even numbers, but cardinality is fundamental to our number concept; Shapiro, Thinking about Mathematics, pp. 133–8.
......[iii] If S and P(S) had the same cardinality, there would be one-to-one mappings from S onto all of P(S). Let M be one such mapping, and let a subset of S, say D, be specified as follows: For each member of S, if the subset that M maps it to contains it then D does not contain it, and otherwise D does. The problem is that since D differs from every subset that M maps the members of S to, D differs from every subset of S, whereas D is by definition a subset of S. That is, D is contradictory, and so there is no such M, which means that S and P(S) do not have the same cardinality. But for each member of S, say m, P(S) contains {m}, and so the cardinality of P(S) is greater than that of S.
......[iv] Omega is the first ordinal number after the natural numbers. Ordinal numbers generalize counting numbers as such beyond the natural numbers (whence their use indexing the Beths).
......[v] Such ordinal addition corresponds to a rearrangement of the natural numbers, e.g. from their natural order (to which omega corresponds) to 2, 3, ..., 1.
......[vi] We could also take unions of Beth-1 sets, since Beth-1 is Actual Infinite; and similarly, Beth-2 sets, etc.

Cantor's Paradox

2010-12-09T07:45:00.006Z

This is the thirteenth of 17 posts, which are collectively Eternity, etc.
......This section is rather mathematical, but we can—indeed, should—begin with the simplest numbers, 1, 2, 3, etc. Mainstream mathematics has axiomatic set theory for a foundation, for such reasons as Cantor’s Paradox [i], and pure mathematicians are certainly free to explore any interesting formal possibilities. But we are primarily interested in possibilities insofar as they are (or might be) grounded in the God that is.
......The natural (or counting) numbers are the products of endlessly reiterating the addition of 1, starting with 1. They are clearly instantiated, because you and I are 2 people. Arithmetic is prima facie the science of such elementary metaphysical possibilities as the possibility of two individuals. Such arithmetical Realism may be difficult to justify atheistically [ii], but we may think of the natural numbers as existing amongst God’s thoughts, arising via His epistemic omnipotence from His perfect grasp of the concept of a thing, whence our informal 1, together with a concept associated with His omnipotence, such as possibility, whence informal addition and its endless reiteration. Note that Realist arithmetic can be discovered a priori under Theism because we instantiate the concept of a thing and were created in God’s image (and we might also be divinely inspired) [iii].
......The endless reiteration of the addition of 1 means that the natural numbers are (collectively) infinite. Many mathematicians have taken them to be Potential Infinite, as Aristotle put it [iv], or as J. S. Mill put it, indefinitely extensible. The addition of 1 is a definite process, but the natural numbers would have no fixed extension if the endless reiteration of the addition of 1 led to growth that could not even in principle be completed. But most of us think of arithmetic as timeless, and the modern view of the natural numbers is that they are (collectively) Actual Infinite, existing as an immutably complete collection, N = {1, 2, 3, …}. Continued...
......Notes:
......[i] Within an axiomatic set theory, Cantor’s Theorem says only that there is no such set of all such sets. For Cantor’s Paradox, see note iii of Divine Attributes.
......[ii] For some well-known problems, see Stewart Shapiro, Thinking about Mathematics: The philosophy of mathematics (Oxford Univ. Press, 2000), pp. 107–289; George Lakoff & Rafael E. Núñez, Where Mathematics Comes From: How the embodied mind brings mathematics into being (Basic Books, 2000), pp. 342–3.
......[iii] For more details, see Christopher Menzel, “God and Mathematical Objects,” in Russell W. Howell & W. James Bradley (eds.), Mathematics in a Postmodern Age: A Christian perspective (Eerdmans, 2001), pp. 65–97 (especially pp. 92–6).
......[iv] To see what Aristotle meant, consider an everlasting fruit-tree. The tree’s endless production of fruit is the ever-incomplete expression of its power to fruit. The total amount of fruit produced is always finite, but always increasing; it is unlimited—is Potential Infinite—because the tree’s power to fruit remains infinite. For more details, see Copan & Craig, Creation out of Nothing, pp. 200–10; Peter Fletcher, “Infinity,” in Dale Jacquette (ed.), Philosophy of Logic (Elsevier, 2007), pp. 523–85.

English Numbers

2010-12-08T02:57:00.006Z

Hartley Slater in The Reasoner 4(12), 175–6, tried to show, from the fact that the number of elements in the empty set is zero, that zero is not, as a matter of English grammar, the empty set—and in general, that numbers are not sets—because we don’t say that the number of elements in the empty set is the empty set. But things aren’t quite that simple.
......To begin with, Slater’s example of zero—which is often defined to be the empty set in pure mathematics—was an unfortunate choice, because mathematicians introduced zero relatively recently. Consequently English remains rather ambivalent about its status. There being no elephants in this room, for example, it’s false that there are a number of elephants here. So from it being true that there are zero elephants here, surface grammar might seem to indicate that zero isn’t even a number (a cardinal number). But zero is of course a number (the number of elephants in this room, the number of elements in the empty set).
......For another example, the numbers one, two, three etc. correspond to the positions first, second, third and so forth. And since no sequence has an element before the first one—that’s what ‘first’ means—so, in that ordinary sense, there’s no zeroth element, and so again, surface grammar seems to indicate that zero isn’t a number (an ordinal number). Nonetheless, there’s a more mathematical sense in which whenever an element is indexed by 0, it’s a zeroth element.
......In many mathematics textbooks there’s a Chapter 0, for example, containing the set-theoretic basics. Of course, such chapters don’t amount to much evidence that mathematicians take numbers to be nothing more than sets. Mathematicians make the standard identification of numbers with sets in order to prove theorems from set-theoretic axioms. They are thereby following in the footsteps of those who did geometry by proving theorems from Euclid’s axioms. And surely few if any geometers thought that there was nothing more to space than Euclid’s axioms. Space was rather the obvious space around us, of which Euclid’s axioms were taken to be true (and obvious enough to be the premises of proofs).
......Now, even if the space that we see around us is Euclidean—having been constructed as such by our brains from our sensory input—it’s surely not unlikely that what Aristotle meant by ‘space’ is non-Euclidean. So, similarly, even if our concept of zero comes (for example) from reifying the definitive property of an absence, it doesn’t follow that it’s impossible that Euler’s ‘0’ referred to an empty set. Indeed, the standard empty set can be an urelement—can be anything that has no members (where membership is an axiomatic primitive)—because its job within standard set theory is simply to have no members, and so in that sense (at least) zero can be an empty set.
......But more to the point, Slater’s argument may beg the question. That’s because if ‘the empty set’ was a definite description of zero then we could say that the number of elements in the empty set is the empty set (for all that we wouldn’t usually). After all, we can say that the number of ones in zero is zero. In general, for natural numbers n, the number of ones in n is n. Perhaps it would be more natural for us to say that two twos are four (for example), and hence that the number of twos in four is two. But such equations all follow from the natural numbers—most obviously those greater than 1—being essentially sums of ones, which seem to be some sort of collection, perhaps not unlike sets of points in that, while their elements are in obvious ways identical, they are distinguished in ways that derive from their origins (as positions in space, in the case of points).
......Two twos are four because any two things plus another two things are four things. And in English, there being a number of things of some kind is just there being some things of that kind. So again, surface grammar indicates that numbers—most obviously those greater than 1—are some sort of collection. And we might expect mathematicians to be the experts on what exactly numbers are. So, since mathematicians prove theorems about numbers from set-theoretic axioms, we’ve some evidence that numbers are sets.
......Still, such evidence is compatible with numbers being axiomatic sets only in a rather abstract way (cf. how the integers with addition are an abelian group). Slater’s argument was based on surface grammar, so it was presumably that numbers are not sets in some more obvious sense. So note that collections in the usual, informal sense can be variable, like a stamp collection, or non-variable, like a chess set. A fundamental question in this area is therefore whether mathematicians have discovered that numbers behave like sets—at least to the extent that the natural numbers are, collectively, non-variable—or whether they’ve just tended to assume that (even though we can’t so easily assume that cardinal or ordinal numbers are non-variable, in light of the famous set-theoretic paradoxes).
......Mathematicians don’t prove the standard Axiom of Infinity—which asserts the existence of a set containing one element for each natural number (amongst other axioms giving such sets the properties one would expect of non-variable collections)—but rather prove theorems from that axiom (along with the others), or work from some other foundation. Philosophical arguments are therefore needed, to assess whether the standard axioms are giving us a scientifically adequate description of the natural numbers or not. But arguments based on surface grammar are unlikely to be of much help in this area. After all, they can’t even show zero to be a number. (For a more apposite sort of argument, see my 2003: ‘Infinite Sequences: Finitist Consequence,’ The British Journal for the Philosophy of Science 54, 591–9, and my 2010: ‘Two Envelopes, two paradoxes,’ The Reasoner 4(5), 74–5.)

Possible Worlds

2010-12-07T03:38:00.005Z

This is the twelfth of 17 posts, which are collectively Eternity, etc.
......As well as all those (correct) statements (see previous post), there are also a lot of truths about other metaphysically possible worlds [i]. Statements of the former kind naturally seem more important than truths about merely possible worlds; and I have not shown that not knowing the former would not make God liable to make mistakes. But balancing the possibility that they do is the possibility that such ignorance is required for our genuine freedom [ii]. And in any case, would it follow from God being maximally knowledgeable—as well as maximally powerful—that He is timeless even if He could only be completely knowledgeable about the future if He was timeless?
......Since the answer is no [iii], let us consider all metaphysically possible worlds, not just this one, under each candidate conception of God. And since God’s omniscience may be less certain than His omnipotence (see section II), let us consider His power as well as His knowledge. And in view of section IV, let us compare libertarian atemporalism with Presentist Open Theism, taking both those conceptions to be prima facie logical possibilities [iv].
......God being possibly timeless means that a world like ours could conceivably be the 4-dimensional creation of a God who transcends its temporal dimension. But it is therefore conceivable that a Presentist God could instantaneously make a 4-dimensional world that is similarly like ours—as it has been so far, and happening also to be that way in the future—but with a fourth dimension substantially unlike Presentist time [v]. So a world as ours would be were God timeless could conceivably be made by Him whether He is timeless or not. And He would be completely knowledgeable about it whether He made it or not.
......Similarly, for any possible spatiotemporal world that a timeless God could make, a Presentist God could conceivably make—and so would know all about—an isomorphic world. But Presentist Open Theism being possible means that this world might have a future that is open, in the sense that there are statements that are neither true nor false but which will be either true or false. And a timeless God could hardly make such a world, because for Him the future has to be completely real.
......Now, atemporalists may not regard that inability as detracting from His omnipotence, whether or not the creator of such a world would be liable to bodge things up in it. But there is also an argument that there are many more metaphysically possible creations if God is able to change (see section VIII), which uses what is shown in the next three posts, that there is no immutably complete totality of all the metaphysically possible whole numbers.
......Notes:
......[i] That is clearly so under Open Theism; and although the deliberate creation of something contingent seems to require several real possibilities to choose between, as well as a single actuality amidst counterfactuals, and hence some sort of change, creation is also taken to be contingent by Mawson, Belief in God, p. 71.
......[ii] Boethius famously argued that our freedom would not be limited by God’s knowledge of what we will be doing were that, not so much foreknowledge, as timeless knowledge (the analogy was with someone knowing what we do, not before we do it, but as we do it). (E.g. see Mawson, “Divine eternity,” pp. 38–40; Sorabji, Time, Creation and the Continuum, pp. 254–6.) Nevertheless, it would still have been true in the past that God knows (timelessly) all about the future. And to see why that might be a problem for libertarian atemporalism, consider a timeless God revealing truths about our future free actions to some of His saints in the remote past (and perhaps even on a distant planet). Is it obvious that our now being responsible for our actions could depend upon Him having done no such thing? (For more details, see Helm, Eternal God, p. 101 ff.)
......[iii] To see why not, consider someone choosing between Open Theism and the hypothesis that she was made by a transcendent computer, which has a complete database on (and complete control over) its creatures’ relatively virtual lives. Only the computer could be completely and infallibly knowledgeable about her whole life. But it would clearly be less knowledgeable (and powerful) than those who might have built an isomorphic computer, and who might have been created by an even more knowledgeable (and powerful) Open God.
......[iv] Cf. Mawson, “Divine eternity,” p. 46 n. 10.
......[v] If this world had been made like that, we should think of Presentism as false, because the analogical—as we should then see it—instant at which God was fully present would then include the past and future. Such a God would be neither everlasting nor timeless (see note vii of Divine Attributes cont.), but is also relatively implausible (see note iii of Omniscience Again cont.).

Bodging Up cont.

2010-12-05T09:43:00.005Z

This is the eleventh of 17 posts, which are collectively Eternity, etc.
......How closely would such interventions as we might expect under Open Theism have to resemble Mawson’s scenario (see previous post)? Why, to begin with, should the world’s aggregate happiness have decreased? The immediate consequence of Adolf’s birth was a little more joy in the Hitler household. And surely God would have intended to intervene further, as necessary to ensure that aggregate’s continued increase, if that had been His motivation. (Making such interventions would hardly cause further problems, as simply making evermore planets or heavens of inherently happy animals or angels might suffice.)
......But in any case, a more plausible motivation for an Open divine intervention would be that aggregate’s eventual perfection [i], e.g. by our becoming a communion of saints. Rather than the Open God intervening to ensure a baby’s birth [ii], He would more plausibly have answered Mrs. Hitler’s prayers in order to help her to relate to Him more fully [iii]. And had He done that, then the satisfaction of His desire would hardly have depended upon how her child grew up. It would have depended upon her free choices—to some extent (He would definitely have so helped her)—but that amounting to luck is generally rejected by libertarians [iv]. Furthermore, even if Mrs. Hitler did not respond by becoming a saint, the Open God could surely try again, and again. And His attempts might become irresistible, as Mrs. Hitler wised up, or perhaps she might become very undeserving.
......So in short, Mawson’s scenario did not show that the satisfaction of the Open God’s most beneficial desires—perhaps that everyone (who is not too undeserving) should end up somewhere heavenly forever—could not be inevitable. So we are left with no reason why we should think of the Open God (of any variety) as a “well-intentioned buffoon”[v], rather than as Jesus [vi], and hence no reason why God should know all about the future (cf. end of section IV). So although there are statements about the future that would not be known by the Presentist Open God despite them being in a sense correct, that sense has not been shown to be significant enough to obstruct divine omniscience.
......Notes:
......[i] Keith Ward, Divine Action (Flame, 1990), pp. 134–9.
......[ii] Swinburne, Is There a God? pp. 114–5.
......[iii] Robert M. Adams, “Theodicy and Divine Intervention,” in Thomas F. Tracy (ed.), The God Who Acts: Philosophical and theological explorations (Penn. State Univ. Press, 1994), pp. 31–40.
......[iv] Timothy O’Connor, “Is It All Just a Matter of Luck?” Philosophical Explorations 10 (2007): 157–61.
......[v] Mawson, “Divine eternity,” p. 48.
......[vi] Richard Swinburne, Was Jesus God? (Oxford Univ. Press, 2008). If Jesus is, like us, a continuant, then it’s hard to see how he could be, not just the signature, but the identity of a timeless God. But clearly an everlasting God could incarnate as fully human, e.g. if we are essentially spirits that produce human minds because we animate human brains.

Bodging Up

2010-12-03T10:22:00.005Z

This is the tenth of 17 posts, which are collectively Eternity, etc.
......Mawson’s main argument had the following preamble [i]. Early in 1936 it was, according to Mawson, vastly improbable that that year’s Man of the Year in Time magazine would be widely regarded ten years later as the most evil man ever. Mawson’s point seemed to be that if a temporal God would have known, early in 1936, that such a change was very unlikely, and if to find something so unlikely is to believe it false, which follows from Swinburne’s definition of “belief” according to Mawson [ii], then such a God would have believed that such would not happen, incorrectly (that man being Adolf Hitler).
......However, while the man in the street of early 1936 may well have found such a change unthinkable, surely the perfectly aware sustainer of the whole world might have known better. And in any case, while we do—and often should—believe things that, upon reflection, we would only regard as highly probable, the beliefs held by the Open God are presumably held infallibly. That latter observation was acknowledged by Mawson [iii], so let us now look at his main argument, which revolved around the following scenario [iv].
......The Open God answers the prayers of Mrs. Hitler by saving her unborn baby from a likely miscarriage, doing so because He wants to increase the aggregate happiness of the world. He does not know that her baby, Adolf, will eventually cause that aggregate to decrease. On the contrary, that Adolf Hitler will eventually produce terrible harm is known by Him to be fantastically unlikely. Still, terrible harm is what happened, and so His intervention did not result in greater happiness. And even if it had, He would just have got lucky.
......Of course if, were God timeless, so evil a man would never have been born, then history would be some evidence that God is not timeless [v]. But the idea behind Mawson’s scenario was that something like it must be possible under Open Theism, because the Open “God cannot will Himself to do anything under a description the truth of which depends on future free actions”[vi]. He can in the case of His own actions, however, via His constancy (see section IV), and in my next post I wonder why He would need to in our case.
......Notes:
......[i] Mawson, “Divine eternity,” p. 45.
......[ii] One’s belief that p amounts to a belief that p is more likely than q for some q, according to Richard Swinburne, Epistemic Justification (Clarendon Press, 2001), pp. 34–6. Swinburne was examining the justification of human beliefs (of that form) however, not describing God’s beliefs (as such). Mawson observed that we tend to believe that a tossed coin won’t land on its edge, even though it might. But the Lottery Paradox—I believe, of each ticket, that it won’t win, so (logically) I should believe that none will, whereas I know that one will—was addressed by Swinburne; ibid, pp. 37–8. And a natural resolution is to describe my belief that it probably won’t win at least that precisely. And in general, logical reasoning seems to involve natural language clarification procedures whose aim is an adequate bivalence. E.g. the ambiguity mentioned in note 31 might be clarified thus: Either something will definitely happen, or else it is, if not impossible, merely possible.
......[iii] Mawson, “Divine eternity,” p. 45.
......[iv] Ibid, p. 47.
......[v] Indeed, a world in which everyone always made good choices is a prima facie logical possibility, so one might wonder why a timeless God would not have created such a world, were He perfectly beneficent. For more on the atemporalist free will defense, see Mawson, Belief in God, pp. 198–271. For more discussion of theodicies, see Omniscience and the Odyssey Theodicy.
......[vi] Mawson, “Divine eternity,” p. 47.

Theistic Presentism again

2010-12-01T10:16:00.005Z

This is the ninth of 17 posts, which are collectively Eternity, etc.
......The collapse view of quantum mechanics (see previous post, also Modern Physical Probability) can help us to understand human agency [i]. But if wave-functions do not collapse then you would now be about to do everything that it was physically possible for you to do [ii], by splitting into lots of you (whereas you are a responsible agent). Furthermore, Perfect Being Theists—unlike most physicists—would not regard the question of what (or who) could collapse the universal wave-function as problematic.
......And our third reason—that there would be nothing to make all truths about the past true under Presentism—is similarly trivial under Perfect Being Theism, because God’s memory might serve as such a truth-maker [iii]. Conversely, Presentism can support Perfect Being Theism. E.g. it allows such Theists to “accept neither the timelessness nor the temporality of the being of God”[iv], insofar as “temporality” means being inside the temporal dimension or being dependent upon temporal constraints [v]. Furthermore, Presentism allows perfect goodness to become omnipresent [vi].
......Presentism is not, then, obviously implausible under Perfect Being Theism. So although there is a sense in which your earlier belief about reading on (or not) may have been correct—perhaps it seemed so with hindsight—it does not follow that an omniscient God would have known what you were going to do. Under Presentism, such a belief would not have described what was originally real—the two possibilities—accurately enough for it to have been part of divine omniscience.
......Nevertheless, it seems to me that if a God who does not know all about how things will be would be liable to bodge things up, as Mawson’s main argument claims [vii], then an omnipotent being would know them.
......Notes:
......[i] Henry P. Stapp, “Quantum Interactive Dualism: An alternative to materialism,” Journal of Consciousness Studies 12 (2005): 43–58.
......[ii] Penrose, The Road to Reality, pp. 783–4, pp. 806–9.
......[iii] Alan R. Rhoda, “Presentism, Truthmakers, and God,” Pacific Philosophical Quarterly 90 (2009): 41–62. Incidentally, even the Presentist past is completely real in the sense that all statements about it have definite truth-values. Indeed, it is at least partially real in the more ontological sense that some continuants that did exist still do.
......[iv] Colin E. Gunton, The Triune Creator: A historical and systematic study (Eerdmans, 1998), p. 92.
......[v] With different senses—e.g. those of Padgett, God, Eternity and the Nature of Time—Presentism would allow us to accept both terms consistently.
......[vi] Paul Copan & William Lane Craig, Creation out of Nothing: A biblical, philosophical, and scientific exploration (Apollos, 2004), p. 162 n. 29.
......[vii] Mawson, “Divine eternity,” pp. 45–9.

Theistic Presentism cont.

2010-11-29T12:07:00.005Z

This is the eighth of 17 posts, which are collectively Eternity, etc.
......The Presentist present is not so much a time as reality. A line with dynamic (and continuous) branches may give Presentists a reasonably accurate picture of how continuants change, but that dimension is only figurative. Nothing real is literally inside it. And while it’s only common sense that we exist within time, we are dependent creatures, currently subject to physical laws in the shadow of death. By laying down our temporal constraints, the Presentist God in a sense created time as we know it, which He transcends. Furthermore He was, is or will be at all such times, and is everywhere immanent. The atemporal God also created time, of course (as a real dimension), and is in a sense spatiotemporally omnipresent [i]. So while there is a sense in which the everlasting God is inside—the timeless God outside—time, it is often more accurate to think of the former as able—the latter not liable—to change.
......As Mawson discussed atemporalist spatiotemporal omnipresence, he mentioned—the second of our three reasons—that “of course scientists are happy to talk of space and time as merely two aspects of a unity, space-time”[ii]. But to begin with, such talk is generally an unreliable guide to ontology [iii]. And in particular, the evidence for relativistic physics is quite compatible with there being an absolute present [iv]. Furthermore, many physicists are unhappy with the picture of time as quasi-spatial, because they take quantum mechanics to be describing the collapses of wave-functions that represent current physical tendencies towards the possible outcomes of such collapses [v]. Such physical possibilities interact as such, even though most won’t have been how things actually were. And that implies the current reality of (what will have been) the non-actual—in the sense of the Actual World of popular (4-Dimensionalist) Possible Worlds semantics—or in other words, that the future is partially unreal.
......Notes:
......[i] Mawson, Belief in God, pp. 48–51.
......[ii] Ibid, p. 51.
......[iii] Similarly, neuroscientists are generally happy to talk of the mind as though it was nothing but an aspect of the brain, and such talk hardly means that we are unlikely to be incarnate spirits if there is a God; ibid, pp. 93–9; Swinburne, Is There a God? pp. 70–94.
......[iv] Richard Swinburne, Space and Time, 2nd edition (Macmillan, 1981), pp. 177–205; William Lane Craig, Time and the Metaphysics of Relativity (Kluwer Academic, 2001); John Polkinghorne, “Time in Physics and Theology,” in Poe & Mattson, What God Knows, pp. 61–74; Dennis Dieks (ed.), The Ontology of Spacetime, Vol. I (Elsevier, 2006); Bourne, A Future for Presentism, pp. 141–203.
......[v] For more details, see Roger Penrose, The Road to Reality: A complete guide to the laws of the universe (Jonathan Cape, 2004), pp. 493–608; Karl R. Popper, The Open Universe: An argument for Indeterminism (Hutchinson, 1982), pp. 87–109.

Theistic Presentism

2010-11-27T12:03:00.005Z

This is the seventh of 17 posts, which are collectively Eternity, etc.
......The concept of time arises as we relate to the regular changes of such continuants as the earth, or a clock. A continuant is anything that is wholly present when it exists—e.g. you may lift up the whole clock—and which continues to be the same thing as its attributes change—you can put the same clock down later. Many of our quotidian beliefs concern such continuants as the ordinary objects around us. So our quotidian beliefs presuppose (and hence imply) something like Presentism [i]. And if you think about it, you can be justifiably certain that you are a continuant. And Presentist Theists take God to be the original continuant. So whether the Presentist Open God changes or not, He is bound to remain the same person, with the same essential attributes. He is not changeable, like a pagan god, but perfectly constant in His absolute power, and boundless love.
......Consequently He might know infallibly that He will—if He has decided that He will—sustain the earth, at least until tomorrow. It is His choice—He has the power not to—but if God determines to stay on some freely chosen course, then there being no real chance of Him deviating from it would surely confirm, rather than compromise, His perfect freedom, His omnipotence. (Cf. the classical distinction between absolute and ordained power.) Surely having the power to do X might cohere with there being no chance of X actually happening. We are, after all, regarding libertarian atemporalism as a prima facie logical possibility. So in short, it may well be false that, under Open Theism, and “given that the world’s having a future at all is dependent on God’s freely choosing to sustain it from moment to moment […] He does not have infallible knowledge of the future of the world in any respect at all”[ii].
......The underlying problem appears to be how easily Presentism is misunderstood. As we try to understand time—to see more clearly what it is like—we very naturally focus upon its quasi-spatial representation, the temporal dimension. Consequently some philosophers think that Presentism faces the dilemma—the first of our three reasons—that the present is either temporally extended, as though earlier and later could possibly be simultaneous, or else it is only an instant, at which no one could be having such sensations as we clearly are having [iii]. In fact, the Presentist present is not extended (except spatially), but enduring. Nor is it thin—as though the future rained down upon the surface of the past but only that surface was real—because it includes the whole world (as a collective continuant) and, for Presentist Theists, the whole being of God.
......Notes:
......[i] Cf. Mawson, “Divine eternity,” p. 41 n. 4.
......[ii] Ibid, p. 38. Here “world” means the whole of creation; Mawson, Belief in God, p. 10.
......[iii] For more details, see H. Scott Hestevold, “Presentism: Through Thick and Thin,” Pacific Philosophical Quarterly 89 (2008): 325–47.

Future Contingents cont.

2010-11-25T11:41:00.007Z

This is the sixth of 17 posts, which are collectively Eternity, etc.
......Ordinary language is vague (see previous post), and the main thing here is that indefinite is not an unreasonable original truth-value under Presentism (given libertarianism), as follows.
......Under Presentism there would have been, before reading on, or not, those two (metaphysical) possibilities for how things could have turned out, not just the one (actual) future. So your belief about the future would not originally have concerned anything definite (insofar as your belief concerned the future directly, rather than via your intentions). The future is not completely definite under Presentism [i]; or in other words, the Presentist future is not completely real, in the sense that not all the statements about it that will ever have been correct are now true.
......As aforementioned, the future might be partially real—be what Mawson calls “real”—under Presentist Open Theism, because then it might be false (see section IV) that “there aren’t any true statements concerning future states of affairs at all”[ii]. However, the reason given by Mawson for the future being real was that some future contingency will either happen or not, and that if “we assume for the sake of argument that [it will] then that’s a fact about the future that someone could in principle have beliefs [about]”[iii]. And while it will either happen or not, as a matter of logic [iv], to presume that one of those two is already a fact is to reject (or ignore) the following picture of Presentist time, as dynamically branching.
......Presentists tend to picture the past as the trunk of a tree whose branches represent the future, one branch for each possible future [v]. If some future contingency does happen—and similarly if it does not—one could in principle look back down the single timeline of the past that includes it happening (or not), and see that fact with hindsight. Further back, and the contingency would definitely be going to happen (or not), but only with hindsight. At the time, it was one of two (metaphysical) possibilities.
......Since Mawson was right that Perfect Being Theists can, quite properly, be agnostic about theories of time [vi], and since the (logical) possibility of Presentism undermines—or at least reduces the significance of—his primary argument, the next section will further describe that possibility, and thereby answer three popular reasons for rejecting it.
......Notes:
......[i] The future is completely definite under 4-Dimensionalism (aka Eternalism); Michael Rea, “Four- Dimensionalism,” in Loux & Zimmerman, The Oxford Handbook of Metaphysics, pp. 246–80.
......[ii] Mawson, “Divine eternity,” p. 37.
......[iii] Ibid.
......[iv] The ordinary meaning of “not” is all the logic we need here. The complicating factor is that, as aforementioned, “X will happen” can mean both that X will definitely happen and that X will, as it turns out, happen. Consequently “Either X will happen or else X won’t happen” is also ambiguous. For more discussion, see Rhoda, “Generic open theism,” p. 231.
......[v] For such pictures, see Denyer, Time, Action and Necessity, p. 12; Bourne, A Future for Presentism, p. 62.
......[vi] Mawson, “Divine eternity,” pp. 40–41.

Future Contingents

2010-11-23T09:30:00.003Z

This is the fifth of 17 posts, which are collectively Eternity, etc.
......Mawson’s primary argument against Open Theism was essentially as follows (the details differ [i]). When you finish reading this paragraph, you will either begin the next one more or less immediately, or else you will do something else for a while. And whichever it is that you actually do, if you now believe that you are going to do it, you would, it seems, be having a true belief about the future. But you might change your mind (you have libertarian freedom), so under Open Theism God should not now be certain that you are going to do it. Might He guess that apparent truth? [ii] But if the Open God knows perfectly well that He is epistemically infallible, then He is certain of whatever He believes. So this kind of argument is essentially that He is not omniscient.
......Nevertheless, it is because you have libertarian freedom that, unless your belief turned out to be false, either you just made it true by continuing to read, or else you made it true a while ago. There is therefore the conceptual possibility, at least, that your belief was originally of indefinite truth-value, neither true nor false (if very likely) [iii]; and that possibility yields one of the two or three commonest varieties of Open Theism (another being Swinburne’s) [iv]. And according to this variety, God can be (necessarily) omniscient, because not knowing something that is not true would clearly not obstruct (necessary) omniscience.
......Now, even if your belief originally had a probability between 0 (false) and 1 (true), “indefinite” might be used to denote all such probabilities (collectively or indiscriminately). So a third truth-value called “indefinite” is essentially what such an indefinite truth-value is. And those defending this variety should perhaps “motivate and defend the denial of bivalence and the attendant departure from standard logic.” [v] So note that there are many reasons for denying bivalence and moving on from our most elementary logic (e.g. see section VII) [vi].
......Swinburne retains bivalence at the cost of omniscience. Such Open Theists think that your belief was originally true or false according to how things turned out. And indeed, had you guessed correctly, your belief would—in that sense—have been correct. Still, had someone originally believed that she would read on, and later thought “I was right,” she might have meant by that only that she had read on (with no thought for how well her belief had originally described what then existed). Furthermore, since she might not have read on, there is also a sense in which her “I will read on” would originally have been false. Nevertheless, although a third variety of Open Theism does call the original truth-value “false” (and so hopes to retain bivalence as well as omniscience) [vii], her “I will read on” need not have meant that she would definitely read on (which was false), rather than that she would happen to read on (which she did), or indeed, that she would probably read on, or had intended to. Such is ordinary language.
......Notes:
......[i] For more details, see Mawson, “Divine eternity,” p. 37, pp. 44–5. For similar arguments, see Helm, Eternal God, p. 67; Swinburne, Is There a God? pp. 7–8.
......[ii] Mawson hoped to show the Open God making mistakes (see section V), but later accepted that He might be epistemically infallible; Mawson, “Divine eternity,” p. 45.
......[iii] For more details, see Elizabeth Barnes & Ross Cameron, “The Open Future: Bivalence, determinism and ontology,” Philosophical Studies 146 (2009): 291–309.
......[iv] For more on this variety, see David Kyle Johnson, “God, fatalism, and temporal ontology,” Religious Studies 45 (2009): 435–54. This variety was distinguished from Swinburne’s by Helm, Eternal God, pp. 109–21. For three varieties, see Rhoda, “Generic open theism,” pp. 229–32.
......[v] Ibid, p. 231.
......[vi] For further reasons, see Grzegorz Malinowski, “Many-valued Logic and its Philosophy,” in Dov M. Gabbay & John Woods (eds.), The Many Valued and Nonmonotonic Turn in Logic (North-Holland, 2007), pp. 13–94.
......[vii] For more details, see Alan R. Rhoda, Gregory A. Boyd & Thomas G. Belt, “Open Theism, Omniscience, and the Nature of the Future,” Faith and Philosophy 23 (2006): 432–59.

Divine Attributes cont.

2010-11-21T11:14:00.013Z

This is the fourth of 17 posts, which are collectively Eternity, etc.
......As aforementioned, God is the ground of metaphysical possibility, if He exists. Let us assume that He does (if only for the sake of agnostic argument). (And let us also assume libertarianism, if only because Mawson’s arguments do). Since God is, presumably, the greatest conceivable being, we can again follow Mawson and say that He is omnipotent, where “an omnipotent being is a being with the most power-granting set of abilities that it is logically possible anyone might have”[i]. It seems to follow that God is infallible, since being able to make mistakes would surely detract from His overall power [ii]. The Open God might be epistemically infallible, for example, in virtue of His beliefs being held by Him only insofar as they are completely justified [iii].
......Does it also follow that God is omniscient (see previous post)? Each possible fact might be associated with a power to know it. But that might be a power to know it eventually. Would it be more power-granting for God to know now all that He will ever know, or for Him to be able to know anything desired as quickly as desired? It’s hard to say, but it seems that the most powerful God is able to increase His knowledge (see section VII), which would stop Him having what Mawson calls “complete omniscience”[iv], but would not obstruct His necessary omniscience (see section VIII).
......What is relatively clear is that God is eternal, i.e. without beginning or end. That is clearly so if God transcends even the possibility of change. But even the Open God is necessarily unending, because there should never be any real (metaphysical) possibility of the ground of all good things not existing. And if the Open God had some primary state, prior to all His other states, we might regard that as time having its beginning in Him, rather than vice versa (see section IV) [v]. So, God is eternal; what is less clear is whether He is timeless or everlasting. Indeed, even that question’s terms can be questioned.
......Mawson’s terms—“atemporal” and “temporal”—were not too bad, being less biased towards atemporalism (aka Eternalism) than were the traditional terms, “eternal” and “sempiternal” (following Boethius). But whereas both “timeless” and “everlasting” connote unfading immortality, “temporal” connotes limitation. E.g. it can also mean civil or secular, as opposed to sacred or spiritual. And crucially (see section IV), the Presentist Open God does not exist inside the sort of temporal dimension that the atemporal God transcends [vi]. To avoid begging the question, and to stay apposite [vii], I suggest that we call God “timeless” if He transcends, not only spacetime, but even the possibility of change, and call Him “everlasting” if He is so involved with us that the future is open (in the sense of Open Theism).
......Notes:
......[i] Mawson, “Divine eternity,” p. 41.
......[ii] For more details, see Mawson, Belief in God, pp. 28–35.
......[iii] For a possible problem with that, see Omniscience and the Odyssey Theodicy.
......[iv] Mawson, “Divine eternity,” p. 37. “A completely omniscient being cannot learn anything,” ibid, p. 41 n. 5.
......[v] See also William Lane Craig, “God, Time, and Eternity,” in Harry Lee Poe & J. Stanley Mattson (eds.), What God Knows: Time, eternity, and divine knowledge (Baylor Univ. Press, 2005), pp. 75–93.
......[vi] See also Alan G. Padgett, God, Eternity and the Nature of Time (St. Martin’s Press, 1992).
......[vii] I am ignoring immutable temporal Deities. And according to my definitions, God would be neither timeless nor everlasting if He was able to change but had already created our future (see note v of Possible Worlds).

Divine Attributes

2010-11-19T09:39:00.005Z

This is the third of 17 posts, which are collectively Eternity, etc.
......I shall, then (see previous post), be arguing that Open Theists can accept God’s necessary omniscience, in the sense of His being bound to know each truth whenever it exists (insofar as truths exist). Richard Swinburne—the Open Theist that Mawson focused upon—disagrees; or rather, Swinburne thinks that because of future contingents (see section III) we should “understand God being omniscient as God knowing at any time all that [it] is logically possible to know at that time”[i]. But let us follow Mawson, and “say that a being is omniscient just if it is the case that for all statements, if a statement is true, then that being knows that it is true”[ii].
......It would be natural for atemporalists to take truth to be timeless, however. (Indeed, even Swinburne seems to do so.) So Mawson may well have meant, by the phrase “if a statement is true” in his definition, if a statement is ever true. But there are many reasons, e.g. Cantor’s Paradox [iii], why truth is not timeless. And whatever the merits of such arguments, our question here is whether God—who may well be the ground of truth—must be timeless. We should therefore avoid begging the question, as we use Mawson’s definition, by not also assuming that statements cannot change their truth-values.
......Of course, Swinburne’s position is, to say the least, not obviously incoherent. So I would not say that the Open God’s omniscience (in our sense) is logically necessary. A logical possibility is primarily a coherent conceptual possibility, logic being the study of correct reasoning. What I will be defending is the logical possibility—indeed, the plausibility—of a variety of Open Theism under which God is bound, of metaphysical necessity, to be omniscient. Metaphysical necessity generalizes the natural (non-Humean) laws of physics so as to allow for such non-physical things as spirits and to allow for miracles and other possible creations (e.g. a new earth) [iv]. And if there is a God, then all metaphysical possibilities would be grounded in Him.
......Notes:
......[i] Swinburne, Is There a God? p. 8.
......[ii] Mawson, Belief in God, p. 35.
......[iii] If there was a set of all the other sets, then via Cantor’s Diagonal Argument (see section VII), there would be more subsets than there are sets; but each subset is a set, so there is no set of all the other sets. That would be paradoxical if “set” meant things referred to collectively. And since for each set there are such truths as what its members are, so there is no set of all truths; Patrick Grim, “There Is No Set of All Truths,” Analysis 44 (1984): 206–8. The result that truth is not timeless follows if “set” is given its informal meaning, of an immutably complete collection. For other arguments to that effect, see Nicholas Denyer, Time, Action and Necessity: A proof of free will (Duckworth, 1981), pp. 79–82; Richard Sorabji, Time, Creation and the Continuum: Theories in antiquity and the early Middle Ages (Duckworth, 1983), pp. 132–5.
......[iv] For more details, see Mawson, Belief in God, pp. 65–7.

Introduction

2010-11-17T10:27:00.012Z

This is the second of 17 posts, which are collectively Eternity, etc.
......In this debate, “God” refers to the perfect person—or Trinity (but nothing impersonal)—who created everything else ex nihilo [i]. If such a God exists, He (they, she) created us, so such Theists naturally think of Him as the greatest conceivable being; and Open Theists are no exception [ii]. Open Theists are so-called because they believe that God keeps some of His options open, as He relates to us [iii]. Such divine openness follows from the sort of freedom (responsibility, creativity) God gave to those made in His image. Open Theists take a libertarian view of free will. And atemporalists—who think that God transcends even the possibility of change—can also take that view [iv], according to Mawson:

On the libertarian view of free will (as it is standardly construed), all that has to be true for you to be free in the future in your choice to do X is that you have the power at that time to do something other than X and on atemporalism you can have this power—have the power to make God either have the atemporal belief that you do X or the atemporal belief that you don’t do X—without having the power to make any belief He actually has false. [v]

The future being what Mawson calls “real” (“that some statements concerning what is now the future are true”) is relatively uncontroversial. Even Presentist Open Theists might think it true that, for example, the earth will still be here tomorrow. “Presentism is the view that only what exists now has any reality”[vi], but what exists now—e.g. an egg falling—could determine how something will be—e.g. the egg breaking—via the laws of nature, and God’s constancy (see section IV), and so make it true now that such will be the case.
......I argue in section IV that Presentism is not implausible under Perfect Being Theism. Mawson’s primary argument—that because of future contingents (see section III), the Open God lacks some infallible knowledge of the future—therefore lacks the significance attached to it by Mawson. Under Presentism, the Open God can be infallibly omniscient (see section III). Presentism can also explain why the Open God is essentially constant, and hence why Mawson should not have thought of the Open God as having no infallible knowledge of the future. And Mawson’s main argument—that by being incompletely omniscient, the Open God is liable to make mistakes—was based on a misunderstanding of Open divine action, according to section V.
......Section VI initiates a more direct comparison of Presentist Open Theism with libertarian atemporalism. Section VII is based on Cantor’s Paradox and is an informal mathematical argument that the numbers of things within possible creations are indefinitely extensible. God’s omnipotence therefore indicates that His knowledge of whole numbers is forever increasing. And I also suggest in section VIII that under Presentism He could, even so, be necessarily omniscient.
......Notes:
......[i] Mawson, Belief in God, pp. 9–27.
......[ii] Richard Swinburne, Is There a God? (Oxford University Press, 1996), pp. 3–19; Alan R. Rhoda, “The Philosophical Case for Open Theism,” Philosophia 35 (2007): 301–11; Alan R. Rhoda, “Generic open theism and some varieties thereof,” Religious Studies 44 (2008): 225–34.
......[iii] For more details, see Charles H. Pinnock, Most Moved Mover: A theology of God’s openness (Baker Academic, 2001).
......[iv] Not all atemporalists do, e.g. see Paul Helm, Eternal God: A study of God without time (Clarendon Press, 1988).
......[v] Mawson, “Divine eternity,” p. 40. For more details, see Timothy O’Connor, “Dualist and Agent-Causal Theories,” in Robert Kane (ed.), The Oxford Handbook of Free Will (Oxford Univ. Press, 2002), pp. 337–55.
......[vi] Rhoda, “Generic open theism,” p. 234 n. 22. For more details, see Thomas M. Crisp, “Presentism,” in Michael J. Loux & Dean W. Zimmerman (eds.), The Oxford Handbook of Metaphysics (Oxford Univ. Press, 2003), pp. 211–245; Craig Bourne, A Future for Presentism (Oxford Univ. Press, 2007).

Eternity, etc.

2010-11-15T10:05:00.043Z

This is the first of 17 posts that, collectively, defend Open Theism against Mawson’s Divine eternity (2008). As I write them, I shall be polishing up what I was writing over the summer, i.e. Eternity, Mawson’s belief and Cantor’s paradox, which was itself the result of a very slow revision of the first half of my 2008 reply to Mawson, Omniscience and the Odyssey Theodicy. The Abstract for the current paper is:

T. J. Mawson believes that the God of Open Theism is liable to bodge things up, by not infallibly knowing all about the future. But I argue that Mawson misconstrued the Open Theist view of divine action. And since the God of Presentist Open Theism ould infallibly know all there is to know, I also argue that Presentism, whose falsity Mawson presupposed, is not implausible if God is the Perfect Being. Furthermore, if God can create arbitrarily many things, then because of Cantor’s Paradox, His knowledge of whole numbers is plausibly growing forever. Yet even such a powerful and hence changing God could, under Presentism, know all the truths of arithmetic. I conclude that the Presentist Open God could be the Perfect Being.

According to T. J. Mawson [i], even those Perfect Being Theists who take a libertarian view of free will should reject Open Theism, so long as they accept what he calls “the reality of the future,” i.e. “that some statements concerning what is now the future are true.” [ii] Mawson argued that because of future contingents, the Open God lacks “infallible knowledge of at least some aspects of the future,” [iii] and that such incomplete omniscience makes “whatever goodness (in the sense of beneficence, not just benevolence) God has a matter of luck.” [iv] I will be uncovering several lacunae in Mawson’s reasoning (in sections III to V), and then arguing more directly (in sections VI to VIII) for my conclusion, that such Theists should not reject Open Theism. The underlying question concerns how God is eternal (sections I and II). The 16 posts will be:
............I Introduction
............II Divine Attributes
............II Divine Attributes cont.
............III Future Contingents
............III Future Contingents cont.
............IV Theistic Presentism
............IV Theistic Presentism cont.
............IV Theistic Presentism again
............V Bodging Up
............V Bodging Up cont.
............VI Possible Worlds
............VII Cantor’s Paradox
............VII Cantor’s Paradox cont.
............VII Cantor’s Paradox again
............VIII Omniscience Again
............VIII Omniscience Again cont.
......Notes:
......[i] T. J. Mawson, “Divine eternity,” International Journal for Philosophy of Religion 64 (2008): 35–50.
......[ii] Ibid, p. 37. Statements are basically possible assertions. For more details, see T. J. Mawson, Belief in God: An introduction to the philosophy of religion (Oxford Univ. Press, 2005), p. 239 n. 6. Questions of truth are essentially questions of how well our words describe the world, of course.
......[iii] Mawson, “Divine eternity,” p. 37.
......[iv] Ibid, p. 49.

The Alternative to Naturalism

2010-11-13T09:00:00.009Z

...is Philosophy, which revolves around debates between Naturalists and Theists. It is a few centuries since Philosophy was the alternative to Theism, but the mainstream of analytical philosophy remains Naturalism, which is essentially atheistic. Science, which Naturalists emphasise, is not atheistic, however, but agnostic. Indeed, the particular sciences are not just agnostic about God, but about most other things too, so that scientists make the most interesting connexions. But Naturalists, no less than Theists, aim not so much to join up the sciences, but to find holes in each others’ arguments.
......As a mathematician, I am quite interested in seeing how mathematics would differ from standard set theory were we created by a Perfect Being. Indeed, I suspect there is an argument from 2 + 2 = 4 to the existence of a Perfect Being who created us. Mainstream philosophy of mathematics takes ‘2 + 2 = 4’ to be what mainstream mathematics—whose foundation is axiomatic set theory—says it is, and aims to develop an atheistic epistemology of set theory. As they do so, their concepts become implausible (whence the possibility of such an argument).
......Ironically, it is because I am essentially a mathematician that my approach to philosophical problems would strike most analytical philosophers as insufficiently mathematical (formal). And incidentally, there is a famous story (quoted from blogcritics) that:

Czarina Catherine the Great of Russia was concerned about the deleterious effect the philosopher Diderot was having on the religious faith of the nobility who were listening to him hold forth on atheism in her court. She encouraged famous mathematician Leonard Euler to confront him, and he did, with the following challenge: “Sir, (a + bn)/z = x, hence God exists—reply!” Diderot, who, according to the story, was completely mystified by all things mathematical, fled the court and
Russia in deep humiliation. Diderot and Euler actually were in Russia at the same time, both at the invitation of the Czarina, but this is a joke at Diderot’s expense that neither Euler the man nor Euler the mathematician would have made. Even if it had been, Diderot—who was actually a fairly capable mathematician himself—would not have been stumped.

Two Contraries of Naturalism

2010-10-30T11:46:00.001+01:00

Naturalists are analytical philosophers who take the natural sciences to be giving us the best foundation for our thoughts. As philosophers, they don’t usually trust scientists to interpret their own discoveries, however, so I wonder what Naturalists contribute to the scientific enterprise. Is it logical rigour and coherence, or just the opposite?
......Naturalists emphasise the very careful observations that scientists make. And indeed, it is important that we be sure of our facts. But when it comes to the most certain thing in the world—that we are conscious entities—they favour materialistic theories that reduce such things away. They may justify their preference on the grounds of ontological parsimony, or overall theoretical elegance, or the fact that natural scientists don’t usually include subjective stuff in their scientific observations. But Naturalists rightly regard as unscientific those who would explain away such less obvious facts as the fossil record on similar grounds (e.g. not needing all those aeons of inhumanity, and there being no mention of such things in a favoured account of reality).
......Naturalists criticise religion for giving too-easy answers. And it is indeed important to get the right answers to the questions the world raises, not just the most convenient ones. But Naturalists like theories that are based on standard logic, which is simple and popular. And standard logic takes a timeless view from nowhere, which is a bit unrealistic. E.g. consider how the Copenhagen interpretation of quantum mechanics, by having an observer-shaped hole in its description of reality, is less tidy but clearly more realistic than those no-collapse interpretations that would have us doing everything it was physically possible for us to do by splitting into lots of different people all the time.

Valid Enough

2010-10-29T11:43:00.001+01:00

The essence of analytical philosophy is the presentation of a valid argument. But often the result is a lot of boring nonsense, many of us find. Why? Well, the reason may be that we are encouraged to work with an absurd definition of ‘valid’. A technically valid argument is, for example, since x and y, therefore x.
......But what about, since the sky is blue, and there’s little wind, I won’t need my umbrella? Technically, that’s invalid because it’s not impossible that it suddenly clouds over and rains. Some philosophers would therefore call it an induction. But I don’t see any generalisation over lots of observations there. And while such a generalisation may well be one of the argument’s many implicit premises, surely it is all the obvious implicit premises that make the argument sufficiently valid for human communication.
......Or, for a more philosophical example, consider Moore’s argument: Since that looks like a tree, therefore that is a tree. Now, we usually make such a deduction subconsciously, but nevertheless, surely such arguments are usually valid enough. And even in more rigorous contexts, how else are we to do science except by taking our readings to be as we read them? What would make such arguments invalid is something like bad lighting, not the mere possibility that we’ve just been taken into the Matrix.
......Indeed, even if we had been, our argument might be valid enough, because we would then be using words in a new external world, and ‘tree’ would usually refer to the new object. Our argument would only be invalidated if we were aware that we were in the Matrix, and if that aspect of our situation was the most apposite.

Definite Enough

2010-10-19T11:56:00.003+01:00

Near enough is good enough in the real world; and in particular, it is good enough for reference, a concept that lies at the heart of philosophical logic. Reference occurs when someone refers someone else to something. Even prior to language, there is a primitive kind of reference that involves being seen to be looking at something, of some commonsensical kind. Now, analytical philosophers usually move quickly from reference to definite descriptions, before getting bogged down in the problem of vagueness. But I would like to suggest that vagueness is not so much a problem at the cutting edge of mathematical logic, as the best way to resolve most problems in philosophical logic. For a very simple example, consider the Lottery paradox: I believe, of each ticket, that it won’t win, so logically I ought to believe the conjunction, that none of the tickets will win, whereas I know that one will win.
......The paradox is resolved if I describe my belief that it very probably won’t win at least that precisely. And similarly, consider the Preface paradox: Each statement in this post is here because I believe it, but I also believe that I have probably made at least one mistake. Also similar is the fact that I believe that what I am now looking at, out my window, is a horse trotting past a tree, even though it might, just possibly, be a painted zebra about to be eaten by an alien stick insect. And what is common to all such paradoxes is that they are most satisfyingly resolved by our clarifying our terms. And quite generally, the power of natural language lies in its flexibility, which derives, I think, from the inherent vagueness of its terms. There are endless examples, so I challenge the reader to come up with a term that could not be replaced by more precise terms if necessary.
......Such subtle vagueness is ubiquitous precisely because our terms are almost always definite enough, so long as we speak carefully enough. And since logic is essentially the study of correct reasoning, it should not ignore the natural-linguistic clarification procedures—such as philosophical analysis itself—that aim at no more than an adequate bivalence. We have a strong bias towards bivalence because as we philosophize we clarify, aiming to maintain an adequate bivalence. But intuitions that logic ought to be bivalent are therefore quite compatible with logic not being perfectly bivalent. After all, questions of truth are essentially questions of how well our words describe the world, and so the logical primitive is not T (true), but True Enough—when we say “that’s true” we usually mean that it’s true enough—and it is implausible that statements are bound to be either true enough or else sufficiently false.

Do Inconsistent Objects Exist?

2010-10-07T16:17:00.011+01:00

Mark Colyvan (2009: ‘Applying Inconsistent Mathematics,’ in Otávio Bueno & Øystein Linnebo, New Waves in Philosophy of Mathematics, Palgrave Macmillan, pp. 160–172), while looking at Inconsistent Mathematics, tentatively suggested that (p. 163): ‘There are times when we ought to believe in inconsistent objects.’
......His example was the infinitesimals of the early calculus, which are widely believed to have been inconsistent. E.g. they were equated to both zero and non-zero quantities (while Newton’s fluxions varied but were inconsistently equated to constants). Nevertheless, the early calculus was widely applied throughout the eighteenth century. So if, as many philosophers assert, ‘we should be committed to the existence of all and only the entities that are indispensible to our best scientific theories’ (p. 162), then it seems that Colyvan’s suggestion makes sense.
......But can we know what our best theories are, without hindsight? Epicycles, for example, were an arbitrarily effective way of coping with real-world ellipses, given a mathematical language of circles. So the astronomy of Copernicus, with his circular orbits and no epicycles, was originally less accurate than Ptolemaic astronomy. But of course, the former was a better theory. It was a step in the right direction. If we can’t, then, know what our best theories are without the benefit of hindsight, then insofar as we now have better theories without such inconsistencies in them, perhaps we should now say that we should not then have believed in such objects. After all, we could always take inconsistencies to be good indications that we need a better theory.
......It is perhaps easier to see that asking if inconsistent objects exist is a bit like asking if impossible things can happen. And of course, if something happens then it must have been possible. Nevertheless, things that seem impossible can happen. And similarly, existing objects may well have descriptions that, while true enough individually (for our usual purposes), are collectively inconsistent (at least on the surface). An apparent inconsistency usually means that our descriptions stand in need of more precision. But it doesn’t mean that they’re too bad to use most of the time. Nor does it mean that the objects so described don’t exist.
......For a ubiquitous example, light as we sense it can be bright (or dull), but photons are dense (or sparse). Our word ‘light’ equivocates between our sensations of light and the light itself. Furthermore, light itself is lots of photons, but it’s also electromagnetic waves, and waves aren’t particles. But it isn’t that light doesn’t exist, of course, and eventually quantum physics described such behaviour consistently enough. It certainly makes sense for us to believe that photons exist (and to be even surer that light exists). And although light also behaves according to relativity physics, which may well be inconsistent with quantum physics, such inconsistency may just be a reason to pursue an even better theory (of light).
......For a more apposite example, the infinitesimals of the early calculus, more precisely described, may be the so-called irreal infinitesimals that were informally introduced in my To Continue with Continuity (pp. 105–107). Suppose there are such continua as I describe in that paper (e.g. space, perhaps). And let x be a real number (e.g. pi), in the sense of an integer (e.g. 3) plus, after the decimal point, an endless sequence of digits in all the decimal places: the first (e.g. 1), the second (4), the third (1) and so forth (59265...). While that isn’t the standard definition of a real number, it’s a definite enough concept, and highly applicable.
......And since the standard axiom of infinity—which says that the natural numbers are collectively a standard set—goes well beyond the Peano axioms, and is rather prone to paradox (e.g. Levy’s paradox), let us further suppose that the natural numbers are collectively indefinitely extensible. If that’s indeed the case, then x is an (infinitesimally) imprecise description of many (infinitesimally) different lengths.
......So if l is a non-zero irreal infinitesimal then x + l = x because, quite generally, infinitesimals are smaller than 1/n for any natural number n, so that adding them to x affects x in none of its decimal places. (And incidentally, the word ‘infinitesimal’ derives from a Latin word that originally meant the infiniteth term in a sequence.) There are, then, consistent (if informal) mathematical objects—irreal infinitesimals—that for the purposes of the early calculus could be adequately described by its descriptions of its infinitesimals.
......So if such continua as do exist are well enough described by such (informal) theories as mine, then surely we could take irreal infinitesimals to be the referents of ‘infinitesimal’ in the early calculus. (Points were not then the same as real numbers, and the natural numbers were widely regarded as indefinitely extensible, so that infinite space would contain infinite lengths and hence geometrical infinitesimals.) Furthermore, insofar as the natural numbers can be said to exist, we could then think of such infinitesimals as existing.
......Of course, if mathematical existence is equated with consistency, relative to some axioms, then inconsistent mathematical objects exist when, and only when, we have inconsistent axioms; and of course, such inconsistent objects shouldn’t exist because they would be too trivial.

Monarchy And Democracy

2010-10-05T12:40:00.004+01:00

Yesterday's Dispatches on Channel 4 was about "the illegal phone hacking carried out at the New of World during the six years Andy Coulson was either Deputy Editor and Editor," of current interest because "by hiring Andy Coulson David Cameron has sanctioned the News of the World culture of impunity." Rupert Murdoch certainly seems to have a worrying amount of influence over our police and politics, centre-left as well as centre-right. The origin of these revelations, in a successful prosecution for hacking a conversation between our two young princes, is of course less interesting. But the consequent importance of the fact that Murdoch could neither buy off nor intimidate the Palace does make me wonder whether the left, and other democrats, should question their traditional Republicanism, in these transitional days.

Ordinary Objects

2010-10-04T14:14:00.009+01:00

Further to the question of whether or not Chairs Exist, I notice that Amie L. Thomasson’s 2007 book Ordinary Objects is out in paperback next month. Basically, it shows how “the claim that there are ordinary objects can form part of a coherent, reflective metaphysical view built up out of our common sense way of looking at the world, in a way that avoids the philosophical problems that have long been feared to plague a common sense ontology” (pp. 7–8).
......The first problem addressed (pp. 9–24) was that if all such things are made of atoms then there’s causal redundancy, e.g. if it’s really atoms arranged stone-wise that break a window (atoms arranged window-wise) then we shouldn’t also have a stone breaking it. But of course, the former just is the stone breaking it, if all such things are made of atoms. Thomasson’s analysis was more detailed, of course, but still readable, and seems to generalize nicely (e.g. the next section addressed the related issue of epiphenomenalism in the theory of mind).

Euler’s ‘2’

2010-10-02T12:53:00.005+01:00

Incidentally, I’ve rewritten two of last month’s posts, and the rewrite—Did Euler’s ‘2’ refer to ZFC’s {Ø, {Ø}}?—will be appearing in next month’s issue of The Reasoner.

True Enough

2010-10-01T01:10:00.006+01:00

What is truth? Clearly it includes (following Aristotle) saying, of what is, that it is, and saying of what isn’t that it isn’t. The obvious contrast is with falsity, with saying of what is that it isn’t, or of what isn’t that it is. Truth, then, is the fit of our words with the world.
......But words exist in a public language and so, given how we come to acquire our linguistic skills, vague meanings are inevitably ubiquitous. Still, we invariably speak within some context, wherein we need only say enough to make our meaning clear enough. Our words can describe the world well enough—they usually do—and then what we say is true enough. And when it isn’t, we can always be more precise.
......We can even introduce new terms into our language, if we have to (as scientists and philosophers). Indeed, there seems to be no logical limit to our ability to be ever more precise. And so to say that something is true is, more precisely, to say that it’s true enough. Bivalent propositional logics—in which each sentence is either true or else false—are just rough approximations to the truth.
......Consider some commonplace examples: The table at which I’m typing this is flat—it isn’t warped or lopsided—but in another sense it isn’t flat, not being perfectly smooth and horizontal. To say that it’s flat is to say something that’s true enough.
......And similarly, to return to the themes of previous posts, ‘grass is green’ is true because ordinary grass (such as fills lawns and pastures) reflects the green bits of daylight (fuzzily delineated bits) ordinarily (e.g. when there’s no drought).
......And it’s insofar as chairs exist that it’s true to say of them that they do.
......And do rainbows exist? Well, in a sense they do (e.g. we can refer each other to them), but there is clearly a sense in which they don’t (much as mirages are not oases).
......What do we mean by ‘grass’ or ‘chair’? Such things form obvious kinds, which is how we come to learn such words. What most of our words have, then, are meanings that are definite enough. Indeed, such vagueness may well be logically necessary, in any possible medium of communication. But even if a more definite language was possible, it’s the vagueness we have which means that our words can be given more definite meanings as required. So a less vague language would in any case be a less useful tool.

Putting the green back in the greenery

2010-09-30T09:54:00.006+01:00

The external world of objective reality clearly contains objects of various kinds, shapes, colours and so forth. E.g. there are conifers (an evergreen). Many students of philosophy (following Locke) learn to distinguish between primary and secondary properties of ordinary objects. The former exist out there, in the objects themselves, e.g. their shapes (and natural kinds). The latter exist only as we perceive such objects, e.g. their colours; and so we soon reach the philosophical problem of perception: We see a tree as green, out there in the world, but the green exists only in our heads (so to speak). The modern scientific picture of the world has it that where we see the green tree are really just various biochemicals, reflecting certain electromagnetic waves toward our eyes. The green seems not really to be where we can clearly see that it is—out there, in the leaves of the tree—but to be only in the pictures of the world that our brains construct.
......Indeed, since such pictures are what we’ve been calling ‘the world’, some might think of the world as in their heads (which is one way to put the green back in the greenery). Many philosophers (following Kant) take the shapes of ordinary objects to be, not primary properties (as Locke thought), but also secondary. The world might really be composed of atoms composed of 10-dimensional strings, for example, with our brains imposing, upon the numerous sensations that come from our sensory organs, the usual 2 and 3-dimensional shapes that we see in the world around us. Indeed, since our brains may even be imposing the basic structure of a number of objects upon our sensory input, some philosophers conclude that ordinary objects just don’t exist in reality (e.g. see Jackie’s comments on my previous post, Chairs Exist). But what do we mean by ‘reality’? Surely we could only mean whatever space it is that includes the people whose language includes such expressions. So there are certainly some objects out there, i.e. other people.
......And similarly, I think, there is a sense in which the greenness that we see really is objectively out there, on the surface of such objects as leaves. That is because we learn the meaning of ‘green’ (as part of learning the concept of colour) by being shown various green objects or pictures (and red ones, etc.) and being told that they are all green (red etc.). Green is therefore something that ordinary objects can be. Basically, something is green if its surface is such that, under normal lighting conditions, it would give rise to the same sort of sensations in those looking at it as they had when they learnt the meaning of ‘green’. So when it comes to something being green—to it being true to say of it that it is green—it is irrelevant what those sensations are, whether they are the same for one person as for another (although they are probably very similar, in view of our similar physiologies); the objective greenness that we see via those subjective sensations is, by definition, less subjective than they are.
......Of course, you usually take the meaning of ‘green’ to be just such sensations as you would call ‘green’, because that is how you learnt to use that word. Indeed, we all do, and so that is also part of the meaning of ‘green’. That equivocation usually goes unnoticed—except in such philosophical contexts as the problem of perception—precisely because it is irrelevant what such sensations are (how they differ between people). And of course, the problem of perception is hardly a mistake of the order of a misperception. How else could we possibly refer to things in an external world, except via our side of our interactions with it? Perception is never a view from nowhere. Even scientific observations are careful perceptions of the external world. And when it comes to predicating existence of something, our most certain knowledge comes from some of us seeing that it is.
......Incidentally, a surprisingly good analogy for the problem of perception is a blind person, e.g. using a white stick to check her picture of where she is. Suppose her stick hits an unexpected obstacle in her path. Just from how her stick reacts to hitting it—how the other end of it feels in her hand—she can tell that it’s a bouncy, light, smoothly rolling object... presumably a child’s ball. She can knock it out of the way and carry on; but in the land of the blind, her word for such bounciness in an external object may well be the same as her word for the way her stick felt in her hand. Nevertheless, she would hardly be tempted to think of the world as full of feelings. It would be full of objects that feel one way with a stick and another to the touch, and in other ways via gloves (or other skin, hair, etc.).

Chairs Exist

2010-09-18T09:44:00.003+01:00

The basic contrast is with imaginary objects: Pixies don’t exist, electrons do; epicycles don’t exist, bicycles do. We learn the meaning of ‘exist’ in a world of tables and chairs, trees and cars, and so when we say that electrons exist we mean that they exist like chairs do. We can spray them onto surfaces, for example, much as we might throw chairs into a van. We can catch chairs and electrons, but not pixies.
......If we doubted that chairs exist, what could we mean by ‘exist’ if we said that electrons exist? That they are in our best theory of reality? But the thing about epicycles is not only that they aren’t fundamental objects, in our best theory. It is that they don’t exist, to be further analysed, and therefore shouldn’t have been in our best theory. Of course, pixies exist within fictions, so they exist fictionally, but that is also to say that they don’t really exist.
......Some philosophers think that chairs are imaginary, that only the atoms that make them up exist, but how could that be right? A chair made of Lego bricks would still be a chair. It would still exist, wholly composed of Lego bricks. Had it been made one brick at a time, with one brick not being a chair, and with no addition of one brick making a chair out of a non-chair, it would exist. Consider how, even though orange fades smoothly into yellow and red, with no sharp boundary, that doesn’t mean that carrots are not orange.

Do Chairs Exist?

2010-09-15T15:01:00.006+01:00

Yes they do:

A physicist will tell me that this armchair is made of vibrations and that
it’s not really here at all. But when Samuel Johnson was asked to prove the
material existence of reality, he just went up to a big stone and kicked it. I’m
with him.

......David Attenborough
No they don't:

Like many philosophers, I don't believe that tables and chairs are
fundamental objects. Like a much smaller number of philosophers, I like to say
that I don't think tables and chairs exist. I have good reasons for my denial.
For instance, it does not appear that there is an exact moment at which a table
comes into existence.

......Alexander Pruss

Euler’s ‘2’ postscript

2010-09-08T01:21:00.001+01:00

Since our reductionists (see previous post) assume that we can refer to abstract objects, let us see if any of the ways in which such reference might occur support their view of the referent of Euler’s ‘2’. It seems not; for suppose, for example, that reference to abstract objects is correctly described by Fictionalism. Then the view in question is like taking most twentieth century utterances of ‘Sherlock Holmes’ to refer to the character recently played by Benedict Cumberbatch on the BBC. Suppose instead that Gödelian platonism is true, so that we have something like a perception of abstract objects. Then the only choice we should make in our reference to them is the choice of their names. Between those two possibilities lies a Full-Blooded platonism, according to which all possible abstract objects exist. But that position is hardly available to those who don’t want—but don’t consider impossible—non-set-theoretic numbers. So in conclusion, it seems that our reductionists are quite eccentric after all.

Euler’s ‘2’ continued

2010-09-07T01:17:00.004+01:00

For a more eccentric analogy (see previous post for previous analogy), let a family of cooks in some dull country be introduced to the meaning of ‘orange’ by means of some imported carrots. Since our cooks want to refer only to ordinary objects, not to such things as properties, which seem to them hardly things at all, they reduce talk of orange things to talk of their carrots. But they would clearly be wrong to take us to be referring to their carrots with our uses of ‘orange’. Indeed, we are not even referring to the colour of their carrots, which might turn yellow.
......In view of the way things are—e.g. the cells of the human retina—a more realistic reduction would reduce orange to the two primary colours red and yellow. And to do something similar for the referent of ‘2’ would take us, not to standard set theory but to psychology. The letters ‘M’ and ‘N’ are angular and black and are, collectively, 2 letters, and it is by means of such examples that we came to know what ‘2’ means. Much as shapes and colours are predicated of ordinary objects, the natural numbers are predicated of finite sets.
......Could such properties be collections? Well, there is a philosophical tradition of reducing properties (e.g. orange) to extensions of properties (the class of all orange things), but there is a well known problem with reducing 2 to the class of all pairs. Set-theoretic paradoxes show that such classes are, if absolutely general (not just the class of pairs of ordinary objects), indefinitely extensible. There is no pre-existing class of all pairs to reduce 2 to. Our reductionists therefore reduce 2 to a particular pair-set. Not being eccentric, they don’t reduce it to something concrete, like a pair of carrots (although that has obvious reductionist benefits), but to something as abstract as numbers are thought to be.
......However, it is not so much a discovery as a technicality to use {Ø, {Ø}} rather than {{Ø}}, or at least, such is the choice not to use some other set, class or category theory, or indeed, constructive mathematics. The set-theoretic axiom of infinity, in particular, is true by definition of all standard sets, but is not so much a discovery as a guess about the natural numbers. Now, that assertion only makes sense insofar as numbers are not sets, but that just means that our reductionists risk losing the ability to assert that the axiom of infinity is only a guess; about what would it be a guess? Such reductionists therefore put themselves in the position of those nineteenth century scientists who, for good reasons, took ‘space’ to mean Euclidean space. Those reasons were just not good enough; and note that various supertasks and other paradoxes currently give us cause to question the truth of the axiom of infinity, construed as a property of the non-set-theoretic natural numbers.

Did Euler’s ‘2’ refer to {Ø, {Ø}}?

2010-09-06T14:03:00.006+01:00

The foundation of mainstream mathematics is standard set theory, within which ‘2’ usually refers to {Ø, {Ø}} (when we are considering the natural numbers, see comments below).
......Alexander Paseau (2009: ‘Reducing Arithmetic to Set Theory,’ in Otávio Bueno & Øystein Linnebo, New Waves in Philosophy of Mathematics, Palgrave Macmillan, pp. 35–55) thinks that those reducing arithmetic to set theory in such a way—perhaps they want their ontology to include only sets, not also non-set-theoretic numbers—may also take most mathematicians, past and present, to have been referring to the standard set {Ø, {Ø}} with their ‘2’s. He (2009: p. 42) made the following analogy: ‘When the ancient Greeks spoke about the sun, they spoke, unknowingly, about a hydrogen-helium star that generates its energy by nuclear fusion.’
......By contrast, anyone taking ‘say, a carrot to be the referent of “2” in Euler’s mouth’ should, he (2009: p. 43) thinks, ‘be an error theorist about Euler’s claims involving “2”,’ and so risk taking too many—according to Hartry Field (2001: Truth and the Absence of Fact, Clarendon Press, p. 214)—of Euler’s words to be untrue. According to Paseau (2009: p. 43), our ‘less eccentric reductionists need not interpret Euler’s arithmetical claims error-theoretically and may respect his intended truth-values.’
......They could, he thinks, take that view even if ‘2’ referring to the standard set {Ø, {Ø}} was not so much a discovery about 2 as a technical convention. As Paul Benacerraf (1965: ‘What Numbers Could Not Be,’ Philosophical Review, 74, pp. 47–73) famously observed, another possible referent is {{Ø}}. So for another analogy, suppose some chromatographers took ‘orange’ to refer to wavelengths of light within some definite range, in order to avoid vagueness and because such a stipulation was sufficient for their scientific needs. They would surely need further reasons to take us to be referring to such wavelengths with our uses of ‘orange’. And similarly, if our less eccentric reductionists have not so much discovered the referent of ‘2’ as accepted a useful technicality, then it seems to me that they should not be taking our ‘2’s—nor Euler’s—to be referring to the standard set {Ø, {Ø}}.

What Rainbows Cannot Be

2010-09-02T12:27:00.000+01:00

Rainbows are clearly not ordinary objects, being more like mirages than oases. But they are just as clearly not fictional objects. One’s conception of a rainbow may well contain false presuppositions, but in that way rainbows do resemble ordinary objects. Consider a red apple. Perhaps what is really there is a 10-dimensional collection of particle-strings within a 4-dimensional block universe, with the red and the spheroid existing only in the minds of certain kinds of potential perceivers of that collection. (That modern scientific hypothesis is not a million miles away from theistic idealism, of course.) Anyway, suppose some fictional meteorologists defined ‘rainbow’ to be a specific sort of event. They might do so because such a technical convention suited their scientific needs better than the rather vague ordinary meaning. And of course, an event is a kind of object (especially in a 4-Dimensionalist world). Now, some relatively arbitrary choices may well have been made as they specified their referent of ‘rainbow’. But that does not mean that rainbows cannot be objects. They are, after all, intentional objects, over which we might quantify. And of course, our intuitions that rainbows are not objects all derive from the fact that they are not ordinary objects. I mention this because it strikes me as similar to Benacerraf’s famous argument about what numbers cannot be.

Modern Physical Probability

2010-05-29T10:34:00.009+01:00

What does 'probability' mean in quantum mechanics? I address that rather philosophical question in Modern Physical Probability, a Google doc that replaces the Geocities pages onto which I had put my MLitt dissertation (all my Geocities links need replacing, I've done this one now because a rewrite of my Two Envelopes post is in this month's issue of The Reasoner:)
The doc has 9 sections:
1. Laplace’s Urn
2. Hájek’s Arguments
3. Von Mises’ Limits
4. Popper’s Propensities
5. Reichenbach’s Limits
6. Mellor’s Personalism
7. Humphreys’ Paradox
8. Eagle’s Arguments
9. Lewis’s Humeanism

Eternity, Mawson's belief and Cantor's paradox

2010-04-30T10:03:00.012+01:00

My answer to Tim Mawson's argument for God being timeless, which I've been writing for two years, is currently the following Google Doc (updated August 18): Eternity, Mawson's belief and Cantor's paradox. I defend Presentism under Anselmian theism, and use Cantor's paradox to argue that God is, if He exists, able to increase His knowledge, and hence is able to change, rather than being timeless (unless logic is even less standard than it is under Presentism); and since I also argue that, even so, He could be omniscient, in the usual sense of knowing all truths, I am also answering one of Patrick Grim's arguments against an omniscient being. And since Mawson's argument was directly against Open theism, given that we have Libertarian (agent-causal) freedom, so I am defending Open theism (specifically the second of Alan Rhoda's three varieties).

Lévy's Paradox

2010-03-23T09:33:00.017Z

You probably learned about the real (or measuring) numbers at school. They are usually written as decimals—e.g. ½ is 0.5000..., and pi is 3.1415..., although most of them have completely random decimal expansions—and the set of them all is the real number line, R. Real number variables are ubiquitous in science, with curves in two dimensions having the form y = f(x); and three-dimensional space—that of our imagination, if not of reality (since it is infinite and Euclidean, rather than Einsteinian)—is usually modeled as R cubed, e.g. via Cartesian coordinates (x, y, z).
......Standard mathematics being the language of science, it would be quite surprising were it very likely to be wrong about the real numbers. Unfortunately scientists usually take that to mean that they can safely assume that standard mathematics is unlikely to be wrong, rather than that they ought to assess its likelihood, and to develop (and use alongside it) the least unlikely alternatives. For the following look at that likelihood we first need some terminology, so let the selection of a number of some kind be completely arbitrary when any number of that kind might be selected, with none being more likely to be selected than any other, and let a Real be a real number between 0 and 1 whose selection was completely arbitrary.
......Our first question is, are Reals plausible? Well, any radioactive particle has a half-life, a period of time such that its chance of decaying in that time is exactly 50%, and so an endless sequence of particles, each followed by such a period, could give us a Real in binary notation (with decays corresponding to 1s, and non-decays to 0s), if we ignore sequences with finitely many decays (since sequences with finitely many non-decays correspond to identical numbers), and if the particles are sufficiently independent (e.g. well spaced out). And such quantities of particles may well exist if space is infinite, or if there are other universes alongside ours (in a multiverse), or if the future is infinite. And physical possibility implies logical possibility, of course, and so Reals do at least seem to be logically possible.
......The problem with that is we therefore get a paradox like that attributed to P. Lévy by F. P. Cantelli (1935, ‘Considérations sur la Convergence dans le Calcul des Probabilités’, Annals de l’Institut Henri Poincaré 5, pp. 1–50). A being limited by little but what is logically possible—say, a god—might know many endless lists of different real numbers, and so he might decide that if two Reals happened to be on the same list, he would use their natural numbered positions on that list as two completely arbitrary natural numbers. He could then write, for each number, a note promising the bearer that many days in paradise, put the notes into two envelopes, and ask someone to take one. That is paradoxical because whichever note she picks the other note was almost bound to have been the better choice because, given any natural number, there are only finitely many natural numbers that are smaller, and infinitely many equally likely natural numbers that are larger.
......The standard resolution of Lévy’s paradox—and similarly, of Freiling’s paradox (and not too dissimilarly, Banach-Tarski’s)—is likely to involve the slightest possible violation of our intuitions about probabilities (and related measures). But what appears like a neat resolution given standard mathematics (assuming there is one) is likely to produce a mess of errors if standard mathematics is incorrect. To have any idea of where those errors are likely to show up—e.g. details of theories of probability (such as Popper's) make a difference to predictions in some high-energy physics (likely to be increasingly applied) and to the relative plausibilities of some theories of mind (and hence to some ethics)—we need, not only such standard resolutions, but also whatever other resolutions and associated theories are not too unlikely a priori. And so we first need to go back to basics.
......The natural (or counting) numbers—1, 2, 3 and so forth—are elementary mathematical entities, defined by the endless reiteration of the addition of the unit, starting with the unit (where the unit corresponds to the elementary metaphysical concept of an individual thing, which is presumed by all logics). It is not such numbers but formal (or axiomatic) sets which give us the standard foundation of mathematics, but even so we are only interested in certain formal structures; and informally, a set is a quasi-spatial (or combinatorial) collection, in the following sense.
......Consider some ordinary objects in a room. That is a set of objects because they coexist together in the same room. And of course, since we have all of them so we have any sub-collection (any subset) of them, coexisting in the same spatial way. To call a collection ‘quasi-spatial’ (‘combinatorial’) is essentially to say that all conceivable sub-collections of it are collections of the same basic kind.
......Now, in our snapshot of those objects, in that room, everything was existing timelessly, and while their subsets were not coexisting quite like the objects were—but were rather overlapping (and were perhaps more abstract)—the subsets were also coexisting timelessly, and so we also have, in the same timeless way, a set of all those subsets—the powerset—of those objects. Natural numbers are certainly rather abstract; and the powerset of the natural numbers has the same cardinality as R.
......By contrast, if some infinite collection (such as the natural numbers) is thought of as always growing, according to some given rule (e.g. an endless reiteration), so that we never have all of its elements—although the finite rule allows us to talk of any of them—then only those sub-collections that could be similarly specified, by a finite rule, would exist in the same kind of way. If the natural numbers are not collectively a set, but are rather as indefinitely extensible as they first appear to us, then most of the standard real numbers do not exist, not as definite numbers, because most of them correspond to completely random decimal expansions, which cannot be finitely described.
......When we first think of the natural numbers, we think of them going on and on forever, so there should be some reason why standard mathematics regards them as a set (and since the use of R is ubiquitous in modern science, it should be a very good reason). Now, we very naturally think of numbers as existing timelessly, insofar as they exist (e.g. as abstractions), but we may also think of mountains as existing timelessly (and similarly languages, human rights and so forth). And such paradoxes as Cantor’s (for cardinal numbers) and Burali-Forti’s (for ordinal numbers) have shown us that, even if the natural numbers do form a set, whole numbers more generally cannot. So even if we find it hard to conceive of the indefinite extensibility of arithmetic (to use Mill’s phrase), that cannot be a good enough reason for us to have presumed so confidently that the natural numbers comprise a set.
......At the heart of Lévy’s paradox is the oddity that every natural number is in roughly the first 0% of the set of all and only the natural numbers. So it would be natural for mathematicians to ask themselves whether the natural numbers go all the way to infinity; and if so, why none of them are anything like infinitely big, and if not then how we could have them all. And a reasonable way for us to think of how they could go all the way to infinity would be to use—following J. Benardete (1964, Infinity: An essay in metaphysics, Clarendon Press, p. 31)—the clear conceptual possibility of three-dimensional space, which could easily contain that many particles, e.g. one every light-year, stretching all the way across infinite space, with each being only a finite distance from anywhere.
......So let us look a little more closely at that answer. Let the first particle (anywhere in space) be particle 1, the next (a light-year away) be particle 2, and so on. It seems plausible that, if we had those aleph-null particles, then particle 1 might move from some place P1 to some other place Q1, and then 2 might move from P2 to Q2, and so on. Such seems logically possible at least; and so we might have all the particles moving one by one (in the given order) from some region P (containing Pn for every natural number n) to some other region Q (containing Qn for all n). Indeed, it seems logically possible that they might do so in such a way that the move from Pn to Qn takes half as long as that from P(n – 1) to Q(n – 1), for all n, e.g. because the particles move faster, or because the distances involved are shorter. And if so then in twice the time it took particle 1 to move from P to Q we will have had all those particles moving, one by one, from P to Q. The number of particles in Q goes from 0 to aleph-null via 1, 2, 3 and so forth.
......That hardly seems paradoxical, and yet if that is possible then it is surely no less plausible that such particles should move from P to Q in reverse order. E.g. if particle 1 had moved between the times of 0 and ½, and particle 2 between ½ and ¾, then we might instead have particle 1 moving between ½ and 1, and particle 2 between ¼ and ½. But in such a way we could go from having nothing in Q at time 0 to having, at any subsequent time, aleph-null things there (and finitely many remaining in P). So upon reflection it seems that having aleph-null particles in three-dimensional space is no more plausible than that we could, by gathering things one at a time, go from having nothing to having infinitely many things without at any time having any other numbers of things than zero or aleph-null.
......Now, the standard view will be that the latter is plausible precisely because it has just been shown how it could be done. But even so, our clear conception of three-dimensional space only indicates the possibility of aleph-null particles if we presume that the natural numbers are not indefinitely extensible; and furthermore, that clear conception actually indicates that they are indefinitely extensible, as we will next see by using—following J. Benardete (1964, Infinity: An essay in metaphysics, p. 149)—the paradox of the Spaceship. But first, regarding that former begging of the question, note that space could be infinite, so that we could travel a light-year, and then another and another, and so on indefinitely, without our being able to travel aleph-null light-years, even in principle, precisely because the sequence 1, 2, 3, and so forth, is indefinitely extensible. The infinitude of such a space—which is what allows us to go any finite distance (relative to some unit of length), and also infinite distances—could not be a standard transfinite infinity, but there are such possibilities. In particular, there is a possibility that I have called ‘C-II’ (2005, To Continue with Continuity, Metaphysica 6, pp. 91–109), in which the infinitude of space could be the reciprocal of an irreal infinitesimal (as could the distance travelled by our Spaceship).
......It seems reasonable to presume that an infinite space—a flat, not an Einsteinian space (and a uniformly smooth space)—is conceptually possible. We standardly think of it as not having parts at infinity—as being isomorphic to R cubed—because, given that the natural numbers are collectively a set, such parts would break that space up into such parts, with gaps (a bit like the gaps in the rational number line) between them, whereas our conception of space is that it is uniformly smooth. Such gaps follow from the gap between the finite and the parts at infinity (which might be reciprocals of hyperreal infinitesimals), and look like 1, 2, 3, …, ..., (such-an-infinity – 3), (that-infinity – 2), (that-infinity – 1), … .
......Even so, there is a conceptual problem with R cubed, because a Spaceship travelling in a straight line, and covering the first light-year in one minute, and then each light-year in half the time it took to go the previous light-year would—were it capable of superluminal speeds (which is conceptually possible)—have vanished or teleported after two minutes. So, insofar as it is plausible that it should not have to vanish or teleport, it is plausible that infinite space should contain parts of space that are infinitely far away from other parts, so that our plausible Spaceship can have somewhere to have gone to. So we have a reason to favor theories that allow such spaces. The conceptual possibility of infinite space (and of our Spaceship) implies most intuitively, not that the natural numbers are a set, but that they are indefinitely extensible (as in the uniformly smooth C-II).
......So, at least one argument for standard mathematics—our intuitively coherent conception of infinite space—has turned into a couple of arguments against it, and if that turns out to be the general rule then the correct resolution of Lévy’s paradox may well be the falsity of standard mathematics. Unfortunately there are surprisingly few arguments for standard mathematics. The main one appears to be that the standard mathematicians cannot all be wrong, but surely the few non-standard mathematicians that there are cannot be wrong about the elements of their profession either; and the problems with using popularity as a measure of metaphysical truth are obvious (given our history; cf. how we could have said a few years ago that bankers could not all be wrong). Those who do not like standard mathematics are far more likely to pursue careers other than pure mathematics, than they are to challenge it from within, unless they are geniuses at pure mathematics (and the numbers of such geniuses may well be evenly divided between standard and non-standard mathematics).
......The final argument that I will consider here is that the main alternative to standard mathematics—constructivism (or Intuitionism)—is obviously unrealistic. So note that there are other ways of looking at the alternatives. E.g. consider how either there is a God, or else there is not. If it is the latter then our evolved concepts of number are unlikely to give us a very accurate picture of how numbers really behave at infinity. But even so we might use—following P. Kitcher (1983, The Nature of Mathematical Knowledge, OUP)—the idea of an ideal mathematician to help us to understand standard mathematics. Which brings us to the former possibility; and the commonest view of God nowadays sees Him as, whilst omnipotent, capable of change.
......Such a God might be endlessly constructing arithmetic, much as He creates, in His omnipotence, all that exists (and arguably commands what is right), doing so forever whether or not standard mathematics is correct, in view of Cantor’s paradox (and Burali-Forti’s). On such a view there is at present some biggest number (finite or transfinite), but by the time we had thought of it existing (although it would be unimaginably huge) God would already have gone far beyond it, in His absolutely objective arithmetic (whence this view satisfies most of the common Platonic intuitions). Anyway, that possibility at least shows that it may well be that most of the problems that people have with constructivism do not actually apply to the most plausible way of thinking of the natural numbers as indefinitely extensible, whatever that happens to be (cf. how long it is taking standard mathematics to find a very plausible proper class theory).
(PS: This post is linked to in the Carnival of Math: Mindmap Edition; and in the 106th Philosophers' Carnival: Philosophical Gourmet; and in the May issue of The Reasoner:)

Two Envelopes

2010-03-03T14:03:00.006Z

Consider, to begin with, a simple sort of Two Envelopes scenario: Mr. E. writes out a cheque for £18, another for £12, and puts each inside an envelope. Showing the two envelopes to his friend, Miss Take, he tells her that they contain money, and asks her to take one as a gift. Miss Take does so, and finds that she has £12. Then Mr. E. tells her that one of the cheques was for 50% more than the other, and asks her if she wants to swap her cheque for the other one. Now, Miss Take knows that either the amount in the other envelope is £12 + £6 = £18, or else it is £8 (since £8 + £4 = £12). So she knows that by swapping she would either gain £6 or lose £4. And she also knows that by swapping she is as likely to gain as to lose, since she has no idea which envelope contained the larger amount. So it seems to her that if she swaps she is 50% likely to gain £6 and 50% likely to lose £4. In other words, the mathematical expectation from swapping is 50% of £6 minus 50% of £4, which is £1. Since that is a positive amount, Miss Take decides to swap.
......She feels justified when she ends up with £18 but of course, she had no real reason to swap because her original choice was blind, and by swapping she is in effect just making that choice again. Still, when she swapped she did have more information about what was in the envelopes, and she seems to many to have had some mathematical reason to swap. Such is the Two Envelopes problem, which has been much discussed following M. Kraitchik’s two wallets (in 1953; for a translation of the relevant passage, see previous link). The discussion usually revolves around the problem of randomly choosing the amounts in the envelopes (with one being twice the other, usually), but it clearly does not matter how Mr. E. came by the amounts of £18 and £12. Maybe it was the 18th of December, and Miss Take’s birthday; or maybe Mr. E. was a fan of Tchaikovsky’s 1812. The main thing was that Miss Take did not know both amounts, and that the envelopes did not reveal which contained the larger amount.
......To give us some perspective on Miss Take’s mistake, Mr. E. then offers an identical pair of envelopes to his colleague, Miss Tree, who also picks the one containing £12. But this time, when he offers her the chance to swap envelopes, he tells her that the square of one third of the larger amount is twelve plus twice the smaller amount (in £s). Miss Tree calculates that either the amount in the other envelope is £18 (since 6 is one third of 18, and 6 squared is 12 plus twice 12), or else it is £2 (since 4 is one third of 12, and 4 squared is 12 plus twice 2). So swapping would either gain her £6, or else it would lose her £10. And for all she knows, a gain is as likely as a loss. So now the mathematical expectation appears to be of a loss, of £2. Even so Miss Tree swaps envelopes, precisely because a gain is as likely as a loss. Having read P. N. Johnson-Laird, Miss Tree knows how easily we can be misled by disjunctions; and she suspects that, since only one disjunct of any true disjunction—such as either a gain of £6 or else a loss of £10—needs to be true, so Mr. E. could, by telling her something true about how the two amounts are related, have led her to calculate practically any mathematical expectation.
......The problem of how we should apply mathematical expectations is neither unimportant nor uninteresting, a clearer way to consider randomly choosing a natural number may therefore be via the following variant of a paradox attributed to P. Levy (by F. P. Cantelli, in 1935), which also resembles Kraitchik’s puzzle: Suppose that a god selects two natural numbers at random—i.e. any of them might be chosen, with each being neither more nor less likely to be chosen than any other—and then writes two notes promising the bearer upon demand that many days in paradise, puts them into two identical envelopes, and asks you to pick an envelope. The puzzle is that, whichever you pick, the other is almost certain to contain a far greater gift. That is because, given any natural number, there are only finitely many natural numbers that are smaller than it, and infinitely many that are bigger. But of course, each envelope is as likely as the other to contain the larger amount. Such a random choice therefore seems to be impossible.
......Paradoxically there are, quite plausibly, random real numbers, if the standard set-theoretic axiom of infinity is not too unrealistic; and that is paradoxical because if a god could choose real numbers at random then, since any countable list of real numbers amounts to a correlation between some real numbers and the natural numbers, so a god who knew several such lists might decide that if two randomly chosen real numbers were on the same list then he would present you with the corresponding two envelopes. Regarding the aforementioned plausibility, note that any particle that could decay is, if considered one half-life into the future, mathematically akin to a perfectly fair coin-toss, whence an endless sequence of such particular half-lives could instantiate a random real number in binary notation (with decays corresponding to 1s and non-decays to 0s); and such sequences plausibly exist if space is infinite, or if it lasts forever, or if there are other universes alongside ours (in a multiverse), and so forth...
(PS: This post is linked to in the Philosophers' Carnival 105:)

Zero's signs

2010-02-22T15:29:00.008Z

Maths begins with 1, 2, 3, and so forth; and a natural next step is to include fractions, and at some point we include 0 and negative numbers. In my last post I wondered if the adjunction of negative numbers should be construed as the introduction of directions. If so then when we include negative numbers we should also be exchanging our original unsigned numbers for explicitly positive numbers (the unsigned amount in the positive direction), but the question then arises of what we should do with 0. There is no numerical difference between negative zero and positive zero, both are just zero, and so we might leave 0 unsigned; but in my 2005 paper I presumed (on page 99) that 0 could be assigned both directions (and that in the case of complex numbers, 0 could have all the infinitely many directions of the complex plane). Intuitions for both views come from the use of 0 to label the origin of geometrical coordinates: To get to 0 from 0 we don't actually go in any direction; but then, to get to 0 from 0 we could travel no distance in any direction. So I'm wondering if there are any good reasons to favour one view over the other (aside from my 2005, which is a reason to favour the latter).
(PS: This post is linked to in the Carnival of Mathematics 63:)

Does Mathematics Need New Directions?

2010-02-09T10:28:00.002Z

I've been wondering recently: For most people, mathematics is primarily the study of numbers. For most mathematicians it seems to be the study of standard axiomatic sets, but let's go with what most people think to begin with (after all, even if we began with sets we would have to explain why some of them are regarded as more important, as standard).
......Perhaps most people would think (with Mill) that the meaning of '2 + 2 = 4' is that two objects plus another two makes four objects. Now, it might be supposed that negative numbers present a problem for that view (e.g. Mark Balaguer thinks it nonsense to think of minus two pebbles being added to minus two pebbles to make minus four pebbles), but think of a tiled floor, with one tile missing, as that is a pretty good picture of minus one tile. And positive numbers of tiles could be represented by tiles placed on top of the tiled floor (cf. holes and electrons in semiconductors).
......Note that positive numbers of tiles are not quite the same as numbers of tiles, in that representation, as there are tiles within the tiled floor. That is not a problem, however, because positive numbers are different from unsigned numbers. The former exist within a mathematical space such as the integers, in which '2 - 4 = -2' makes sense (where +2 and +4 have, as usual, been written as '2' and '4' for convenience, because of the obvious isometry), whereas the latter do not, of course, allow you to take 4 away if you only have 2 to start with.
......Now, when we extend the negative numbers to include negative lengths (as we extend whole numbers to include fractions and other lengths), a better representation would have sea-level in place of the tiled floor, and consider extending positive heights of land above sea-level to include depths (negative heights) of the sea-floor. Or we might, of course, look at money, with negative money representing either a flow of money out of an account or, within the account, a debt. And we might, more generally, think of 'positive' and 'negative' as the names of two directions.
......That also fits nicely with the extension of the real number line to the complex number plane, in which there are not just two but infinitely many directions. Historically, the complex numbers (e.g. the square roots of minus one) were only taken seriously when mathematicians could picture them as extra directions. And now we find the complex numbers being as applicable in science (e.g. in quantum mechanics, the foundation of chemistry, electronics, lazers and so forth) as the negative numbers are in everyday situations, which indicates that we are here cutting nature at its joints, so to speak.
......Although Balaguer thinks we must use sets to get the negative (and similarly, the real) numbers, surely we use a concept of direction to make sense of, e.g., 2 - 4 (and similarly, of the square roots of minus one), and surely that also accounts for how useful we find such numbers, as we make our ways in the world. (And incidentally, the real numbers show how our number concepts might be judged empirically, as Mill thought.) In other words, signed numbers and complex numbers may best be thought of as elementary vectors (another fundamental mathematical concept with many applications).
......And so the question of what we mean by 'direction' arises. A very ordinary use of 'direction' is when one gives someone directions to get somewhere. When you get to the next crossroads, take the first left, for example. Note that the road to go down is picked out by a direction relative to where you will be when you have to go that way. And roads can be travelled in two directions (so I am not here thinking of Fregean directions, which are sets of parallel lines), and so I am wondering how deeply the concept of a direction is related to the concept of an option.
......Were there no choices, there would just be all those roads, where they are (much as sets are abstract objects that just sit there in the Platonic realm, so to speak), and little sense to logic. So we might expect choice to be a fundamental mathematical concept. Certainly the word 'choice' appears in the foundations of mathematics, there being an axiom of choice in standard set theory, and choice sequences replacing real numbers in constructive mathematics. But more deeply, our mathematical concepts are likely to be grounded in mathematical intuitions that we share with other intelligent animals (e.g. via the number sense), and intelligent animals make choices (of course).
(PS: This post is linked to in the Carnival of the Animals:)

From the Incarnation to the Trinity

2010-01-21T10:37:00.004Z

For monotheists, the idea of the Trinity (the one God being three individuals) can seem like polytheism. A popular model of the Trinity regards each person (Father, Son and Holy Spirit) as a relation between the other two, but not only is that quite mysterious, it hardly answers the objection because either such relations are distinct (because what is related is) or else they are not. Still, God is presumably more real than the mundane world, if He created it, and it may seem that our relationships are what make us truly real, more than mere things. And if we think of the Creator of the world as to us a bit like a dreamer is to his dreams, then again we have the sense of Him as more real than we are. And furthermore, even the Trinity may then come to seem less objectionable.
......Let us assume that there is a God who created us and the world around us ex nihilo. Such a Being (the ground of being) might be apprehended philosophically, although to many even this sparse conception of God seems paradoxical. Many have preferred to think of creation in terms of emanations from God's being, or of the forming of an energy (formless matter) that coexisted with God from eternity. But maybe God is, to the world, not too different from how we are to our daydreams. Our dreams are hardly real, but the thought is not that the world is just a dream, but that as we are to our dreams so God is, in some ways, to us (and the material world). God is, then, more real than what we ordinarily think of as real (the obvious objectivity of our world deriving from its dependence upon Him). And then we do not have to think of how mere matter could give rise, in some arrangement, to mind, because the original Being was Spirit.
......Let us add to that rather philosophical picture the idea that God became a man, Jesus. Many religions have stories of avatars of gods or goddesses, or even of God, and the divinity of Jesus did occur to the early Christians for some reason. But again, God's incarnation seems impossible (or blasphemous) to many. Still, it is quite reasonable to think of ourselves as spirits in a material world (and it is up to God what He does with His creation). That dualistic approach to psychology is unpopular amongst scientists at present, but modern science actually supports it (and there is an underlying monism for theists) because chemistry being fundamentally quantum mechanics all but solves the old problem of how spirit could interact with a human brain to give us our human minds. So, if God is also Spirit (a spirit more real than we are) then it is not unthinkable that He might similarly incarnate, giving Him a human mind (and body).
......A problem with the idea of God incarnating is how He sustains the universe while He is wandering around as Jesus. But how does the Incarnation look if God is (in some ways) to the world as we are to dreams? Well, dreams come in a great variety, but some end in something more like a daydream, in that the dreamer can deliberately alter them as she wakes up. Such dreams can be very vivid, so they seem very real to oneself, inside their world (so to speak) to begin with. The self in such a dream is at home in the dream, and can even be quite unlike the waking self. But one might surprise oneself in a nightmarish dream, for example, by falling and then finding that one can fly. In later dreams, flying might seem a realistic option, within the dream (whereas other daydreamish possibilities might not). But the more vivid one's control over one's dream, the less realistic the dream seems, and the more one wakes up. There seems to be a play-off in our dreams, between losing oneself in the dream (it seeming real) and one dreaming of whatever one wants (it being like a daydream).
......Now, God is presumably not much like us (or any created thing), but we do have something like a threefold aspect with respect to our dreaming. There is the theme of the dream, which we may have more control over as we wake up (cf. the Father), the dreamer aware of herself as lost in her dream (cf. the Son), and the person one is when awake or daydreaming (cf. the Holy Spirit). Since dreams naturally happen to us, the theme is naturally an unconscious aspect of our dreams, as is the mechanics of how we dream. And unlike God we do not create other people by dreaming. But if God incarnating as Jesus is a bit like our being in our dreams, with something like that sense of two different selves, then the difference that creation involves created people like us could plausibly be associated with what we would perceive as the glory of the Holy, the paradoxical presence in creation of the transcendent ground of being, revealing Himself to us as directly as any of our perceptions could be. It seems that it is by such theophanies that prophets become aware of the God of Abraham, and in similarly direct ways that we become aware of the reality of the Holy Spirit amongst us. We may then wonder how this more real than real personal presence could be related to the creator of the world, the ground of all mundane beings, and to the life and history of the works of Jesus.
......Still, this is only a vestige of the Trinity (if the Trinity is real), not a good model of it. E.g. it hardly helps us to think of the one God being three selves simulataneously. But these Selves are not like created selves. And one can become aware, as one wakes, of the three aspects to one's identity when dreaming being all oneself and being all coexisting, even if one can only do so by switching one's agency and awareness between them. And since that is not like acting in three different ways, not like one's character going through three stages as one grows up, but like being in three different ways, in relation to the same dream, and since this is only an analogy, so one might see how the one transcendent Creator ex nihilo might be, in relation to His creation, what we would naturally perceive as three people. Of course, to see that possibility one must examine such experiences as one wakes up, and then think about them analogically (under the assumption that God has made us in His image and incarnated with us and is amongst us now).
......There can be a moment (which can be protracted into a series of moments) as one wakes when one is not aware of oneself in bed, in the world, but one is aware of the dream as a dream, when one can go back into the dream and continue with it, or change the dream, and go back into it, or think about why the dream was as it was. If dreams were not something that happens to us, but we were more in control of the dreaming process (as we are when daydreaming, or when walking around the real world), as God is presumably in control of His creation, then one would also at such moments be aware of oneself behind the unconscious aspects of dreaming, the creation of the dream landscape and the other characters. So if those other characters were as real as oneself (in the dream), and if one was not unconscious of much of the dreaming process, there would be something like a threefold structure to such moments, corresponding very roughly to the Trinity.
......As creatures we never relate to a God who is not relating to His creation, so it is highly speculative how He would be without creation. But presumably He would have begun then with His responsible ability to create other spirits (centres of awareness and action), and so perhaps with a Trinitarian structure (cf. one's orientation upon waking). Now, realistic thoughts about the revealed God (Jesus Christ and Holy Spirit) are difficult enough, and thoughts about the Trinity as it is in itself are plausibly beyond us, even if the Trinity has been revealed to us in history. But it is at least possible for us to see how we do not have to think of the Trinity as merely how the One appears to His creatures as He reveals Himself to them (to us).
......Suppose that we exist in a 2-dimensional world, Flatland, e.g. as thoughtful triangles, and that transcending our world is a 3-dimensional object, a cylinder. As it shows itself to us by passing through Flatland, it might appear to us as a circle suddenly appearing and disappearing, or as a rectangle slowly appearing from and disappearing into a line. As triangles we would naturally think of the reality as a circle or a rectangle, appearing from nowhere. But the reality is more than that, and the claim that the circle and the rectangle were the same being is not the claim that a circle can be square. And nor is it the case (fictionally) that the being is two shapes (like the plan of a cylinder), or an intrinsically shapeless thing that can take on the form of any shape as it appears to us.
......I don't think that either analogy, the cylinder or the dream, will yield a very accurate model of the Trinity, but they may help us to see that the Trinity might be realistic, by resolving some of the paradoxes we find with other analogies (e.g. the relational Trinity). I'll have to think about it some more (and I'd be glad of your thoughts). The world is not a dream; and God is not much like us, whether we are awake or dreaming. But all our thought about the world involves analogical reasoning, and we might expect that a good grasp of transcendent truth would involve even more of it. If God really incarnated as Jesus, why should that not give us a Trinitarian view of God? And why should God not incarnate in His world? Why should He not create people (if He could) to whom He could reveal Himself like that? Even the creation of a pebble ex nihilo can seem impossible to us, but to God it is plausibly as possible as a daydream of the seaside.
(PS: This post is linked to in the Christian Carnival CCCXII:)

A Stab at a Dogma

2010-01-15T09:55:00.003Z

The Trinity, one God being three who relate to each other, seems paradoxical, prima facie, but perhaps it is only realistic (cf. how chemistry, as it became more realistic than alchemy, turned into a far stranger quantum mechanics). Let us presume that God created us, and the world around us, ex nihilo. Could He have incarnated as one of us? Well, He is presumably omnipotent; and perhaps we are essentially spirits, currently limited by our incarnation in human brains (with which we interact quantum mechanically), and perhaps God is also Spirit (who made us in His image).
......And an obvious way for us to think of creation ex nihilo is by analogy with the way we dream. When one dreams there is the creator of the dream (i.e. oneself), and the character in the dream whose point of view one has (i.e. oneself), and all the other stuff, which is not real but which corresponds (in God's creation) to us and the world around us. This is not supposed to be a very accurate model of divine incarnation; but if the other characters in one's dream were aware (as they obviously could not be) then they would naturally perceive one as a character with special powers and centrality, identified with and yet different to the creator of everything in the dream (including that special character's appearance), most strangely including themselves (which is where the analogy most obviously breaks down, but which may be where the Holy Ghost comes in).
......And quite generally perception (e.g. of a tree) seems to involve phenomena (e.g. green leaves) that are objectified (as what 'green leaves' refers to) in a rather paradoxical way (i.e. the problem of perception). Even when it is a perception of other people, so that we have a relatively direct knowledge of the kind of object, there is still an obvious distinction between how they seem to us and who they really are. And note that if there is a God then Idealism is not especially unrealistic. The connection between how something looks and what it is really like is made on the basis of wide experience and wise conjecture; and if the creator of everything else ex nihilo did incarnate as one of us, He might well be perceived by us as something more like a Trinity than not.

Not-So-Free Thinking

2010-01-14T11:27:00.012Z

Enlightened critics of religion like to point to a history of intolerance of criticism in religion. So it is nice to see Terry Eagleton and Karen Armstrong in fifth place in the New Humanist's 2009 Bad Faith awards, with about six-and-a-half percent of the votes. The New Humanist's article points out "that both have written books this year criticising the New Atheists and mounting what some might call a more sophisticated defence of religion." Quite generally it seems that whenever people talk about important things, there will be those who find good criticism most irritating; and the more popular the philosophy, the more politics there will be.

Duck Soup

2010-01-13T09:38:00.002Z

A plume flies up and plummets
back to flatness. A ponderous pause
and a duckling bobs fluffily up.

Others submerge with further spurts
and splurge, as busy as bees
in the water-lillies. Their bug-like bills
sip duck-poo soup or snap at flies
like aquatic spiders with bedraggled butts.

A lardy mallard waddles by,
his green dragonhead evoking
Vikings as broadsides splash
beneath his lobstrous feet.

Weary of malarkey, of fuss and flap,
his butt fans out like a pack of cards
and onto a flagstone flops. His wing,
sporting a flattering lilac quadrilateral,
flickers and flings off an oddly fluttering
splodge of soggy leaf-litter.

Wary of injury, of malady,
but not too flustered he gingerly
stretches out his white-collared neck
for sumptuous croutons,
a bit presumptuously.

Unhinging winds fringe
maroon-fingered moon,
like a waiter with a supernatural soupspoon,
a crater of rubble like a burst bubble serving
as a soporific seat of tranquillity
for a duck quacking up a melody
of sounds pacific:

Talk about a duck
floating on a lake,
looking like a wooden decoy does.
Talk about a drake
ducking wooden ducks,
making all the ducklings he can make.

Jung, gifted and read

2009-12-11T12:28:00.004Z

...but not read enough, I think; sadly, his Red Book is unlikely to change that.

Deep Thought

2009-12-10T14:01:00.019Z

First things first.
......And yet I find 'zeroth' in my dictionary, to refer to the one before the first one. In Wikipedia I find: The zeroth item is the initial item of a zero-based sequence (that is, a sequence which is numbered beginning from zero rather than one), and surely 'first' would have done just as well as 'initial' there. In such a sequence, the first element is also the second; that is, a certain equivocation has been introduced, a new sense of "first" added to our initial, informal sense.
......The idea of an ordinal number zero comes from the ordering of the integers on a number line, from negative to positive infinity (exclusive). But when we use ordinal numbers to include negatives, as with years, we naturally make the direction more explicit (e.g. with BC and AD) and exclude year zero (whence all the fuss about when the millenium began).
......Modern maths does not like directions. It finds it best to begin with a collection of natural numbers {0, 1, 2, 3, ...} that are most fundamentally ordinal numbers, and are usually reduced to pure sets. Mathematics does not like directions. Following Euclid's reduction of geometry to logic, geometry was naturally reduced to analysis following Descartes, and then arithmetized, with an arithmetic reduced to set theory.
......And yet mathematicians do like directions. Imaginary numbers were only taken seriously as numbers (like the negative numbers) when it was realised that the positive and negative imaginary numbers i and -i could be regarded as unit distances (like 1 and -1), specifically in the direstions perpendicular to the positive and negative directions of the "real" number line in a "complex" plain (which is how complex numbers are introduced to students).
......Given the foundation of the concept of direction in our experience of space, I would guess that if there really are extra dimensions in physics, they are more likely to correspond to the complex numbers of quantum mechanics (e.g. by representing dimensions of actual physical possibility) than the six dimensions of phase space (nor the ten or twelve of string theory, etc.).

Eternity

2009-10-06T12:32:00.007+01:00

Following on from a paper I've been writing for the past year (see previous posts under 'Theology'), the following MTh essay finds no good answers to the question Why should Christians believe that God is timeless? The essay glances at reasons given by Boethius, Helm and Mawson (and so forth), so I'm wondering what (if anything) I've overlooked?

Unretiring enigMan

2009-10-01T12:37:00.007+01:00

Hi; tired of being retiring, I stretch my fingers with intermittent posts (on-going) and, as our droughtless summer (of many Painted Ladies) draws slowly to a close, I think its cultural highlights were: a sci-fi film of the good old 'new wave' kind, Precinct 9; and, more spook story than sci-fi, Iain Bank's Transition; and 60's nostalgiac Inherent Vice; and 90's nostalgiac Menage.

Enigman at the Crossroads

2009-05-02T10:20:00.014+01:00

Tiring of blogging (not so much reading others, as posting my own), it's time for Enigman to retire. There's a local conference in a couple of months, but even if that inspires me with any interesting thoughts, they'd be hopeless for blogging purposes. It seems to take me months to make my thoughts coherent. Incidentally, a lot of my links will stop working soon, as my web-pages are on Geocities, which is closing this summer. I may move those web-pages to this site, probably after re-writing them (most were written a few years ago). The other thing I'll be doing this summer is starting to study theology properly (and catch up on my reading). My philosophical interests were re-ignited ten years ago, via the metaphysics of mathematics, but were nearly quenched by professional analytical philosophy (with few exceptions, e.g. Iris Murdoch). Anyway, thanks for the comments, which made trying my hand at blogging worthwhile.

Eggheads

2009-03-03T15:00:00.006Z

"When I use a word," Humpty Dumpty said in a rather scornful tone, "it means just what I choose it to mean—neither more nor less."

Lewis Carroll's words (from Through the Looking Glass) sprung to mind as I read Slater's response, in this month's The Reasoner, to my reply to his response to my "Liars, Divine Liars, and Semantics." (Basically, when I say something to you I could mispronounce what I meant to say, and you could mishear it, and so the truth of what I actually say must be, as a rule, independent of what you or I happen to think I said.) Equally trivially, Thursday's afternoon play on Radio 4 is The State of the Art, and on Friday the Manhatter's teaparty hits the big screen.

The Problem of Knowledge

2009-01-29T14:56:00.006Z

more animals

A Linguistic Puzzle

2009-01-29T09:49:00.004Z

Say you read something meaningful—e.g. ‘my tabletop is flat’—where’s the meaning? A fairly standard answer is that in that case, it’s a proposition whose constituents are the tabletop and something like the property of, or the extension of, being flat. That is, the meaning is in the flatness of the tabletop. A philosophical problem is to connect my tabletop and these words, these patterns of light and dark. All you get from me is patterns of light and dark. Never mind how I came to produce them. Unless they’re like words in a book of magic spells, all they are is stuff in the world exhibiting such patterns. Any meaning they have for you must be put into them by you. The paradox is that you read words to get from them someone else’s meanings. You often know, on reading something really meaningful—e.g. a great novel—that you’re not just getting back what you put into reading the words, not just a rearrangement of what (since you can read the words) you already knew. That’s the paradox; the puzzle is how analytic philosophy hopes to resolve such matters.

Our Lewisian paradise

2009-01-19T14:30:00.005Z

So-called ‘modern’ philosophy began with Bacon and Descartes, but it all looks pretty dated (if sometimes deep) up until around 1973, when David Lewis gave analytical philosophy the standard PW (possible worlds) analysis of counterfactual truth. A comparable event might be the giving by Cantor (according to Hilbert) of a set-theoretical paradise to standard maths (which is less than ideal if one is applying the maths in theoretical physics, I think). What PW talk gives philosophers, it seems to me, is a new way to argue fallaciously, rivalled only by the use of statistics in politics. But anyway, here is Lowe’s (2006: 12) succinct description of Lewis’s analysis:

A counterfactual of the form ‘If it were the case that p, then it would be the case that q’ is said to be true if and only if, in the closest possible world in which p is the case, q is also the case – where the ‘closest’ possible world in question is the one in which p is the case but otherwise differs minimally from the actual world.

Lowe followed that with what would, until relatively recently, have been a stunningly fallacious argument against mental physicalism, all wrapped up in PW talk: a gift to any intelligent physicalist, who’s thence able to refute an objection to her position that comes with all the modern trappings of the authority of modern philosophy. But to step back a bit, what’s wrong with Lewis’s analysis? To begin with, subjunctive talk equivocates like anything. And then there’s the problem of saying what is, in the relevant way, possible; a problem Lewis solved in an implausibly Humean way, which was at least elegant in a principled way, if evidently false. And of course, what is to count as ‘close’?
......Suppose I’m trying to tell you something, and I know (since I know what it is) that you’ll find it hard to understand what I’m saying. I might say, ‘If you knew what I was trying to tell you, you’d know how difficult this is.’ But of course, if you did know then it would be trivially easy to tell you about it. Perhaps I meant that after I’d told you, and you’ve understood me, you’d agree that it was difficult to tell you. But suppose it’s so hard to tell you that you never do get it. Does my meaning really depend upon which of the possible worlds in which I’ve told you is most like the actual world, in which you didn’t get it?
......What if the difference is just a few neurones that you were born with, for example, but that those neurones also make it hard for you understand why it was difficult to tell you (since you then find such things so obvious)? What if lots of things; and so basically, how could all that really affect the meaning—and hence the truth—of what I’m saying? After all, we do seem to have got bogged down in an awful lot of fallacious arguments and counter-argumetns since the 1970s; which may be ideal for professional philosophers in a stupid economy, but less so for those applying logic to the real world.

Cartesian dualism, ii

2009-01-16T10:23:00.002Z

I’ve yet to find a good philosophical argument against such substantial dualisms as (for the commonest) that our psychology results from the interaction of spiritual souls with the physical brains in which (so to speak) they’re incarnated. The two commonest arguments are (i) asserting the closure of the physical and (ii) failing to see how the spiritual could interact with the physical. Both are clearly fallacious as I’ve stated them, but I’ve yet to find a substantially fuller, non-fallacious expression of either. Now, I’ve blogged on (i) already, and have little to say about either anyway, but I’ve just been reading Lowe (Erkenntnis 65, 5–23), who put (ii) as follows (2006: 7, 11):

[...] according to Descartes, whereas the mind has beliefs, desires, and volitions, but no shape, size, or velocity, the body has shape, size, and velocity, but no beliefs, desires, or volitions. [...] it is often complained that it is completely mysterious how an unextended, non-physical substance could have any causal impact upon the body – the presumption being, perhaps, that any cause of a physical event must either be located where that event is, or at least be related to it by a chain of events connecting the location of the cause to the location of the effect.

As put, the problem seems to be one of mere conceptual possibility, which is easily answered. By typing into your keyboard you can make virtual beings move about in cyberspace. Clearly you don’t have to be where they are, in cyberspace, to be able to move them about. So it isn’t so very mysterious how such things are possible. And even if it were, why presume that would be a problem for dualism, rather than a personal failing?
......As Lowe notes, people said that Newtonian action-at-a-distance was completely mysterious, and maybe it was, and is, but there was hardly any argument there against Newtonian physics (except in the minds of some philosophers). The truth turned out to be far weirder again, and it was to be had by working through Newtonian physics. There is that other problem, of how exactly the interaction works, but the way towards answering that is the relatively hard way of science, and why should it not go through Cartesian dualism?
......My analogy only worked because of the causal link between your fingers moving on the keyboard and the consequent virtual motion (as expressed in actual space on the screen), which goes via continuous paths in space (if we include force-fields in our ontology), but still, it did work. It suggests that a possible Cartesian response is to give the body, not only a spatial location, but also another, non-spatial location, at which the soul acts. How plausible is that? In the natural theistic context of Cartesian dualism, it’s very plausible, since God created space, and is himself located elsewhere.
......And suppose that Cartesian dualism is false. Then there’s some other true theory of mind. Somehow the physical brain, which changes its form and its atomic constituents continually, is associated with a subjective unit (the mind, which we know directly), which is continuously the same person. So if there could be a non-Cartesian theory, then there’s some way of associating with the physical brain a unique continuant of some sort. It is only to that that the Cartesian theory has to associate a soul. And a very simple and natural (in the Cartesian context) way to do that would be by divine stipulation, God associating each such brain-correlate with a unique soul.
......In many ways that’s far simpler and more natural than the sort of Humean regularity approach to scientific laws that philosophers are often led to by considering how mysterious are nomological necessities (a consideration that most scientists rightly ignore). If souls are possible, then they would have individual existences, in some logical space (say heaven), and would interact in some way (say via spiritual bodies). And if so then matter would’ve been created to be such as could be used in such ways (for some reason). The details are for scientific discovery, but the mere possibility is not really so mysterious.

What's a mathematician?

2009-01-14T14:23:00.009Z

A mathematician is a device for turning coffee into theorems, according to Renyi's famous joke. But since most mathematicians don't prove theorems nowadays, I tend to think of the natural mathematicians as those finding More or Less as gripping as an Agatha Christie mystery (whether or not they like set theory).
......On the news this morning, there was a story about researchers at Durham have discovered that coffee makes us hear voices. They conceded that maybe those who hear voices tend to be more stressed, and so drink more coffee, but that seems like an odd concession to me, as people who're stressed usually turn to, if not alcoholic beverages, then chocolate or cigarettes, or cannabis.
......Scientists have also claimed that cannabis can trigger psychosis. That seems more plausible, as cannabis is the famous hippy drug, but I wonder even about that. Many of the most obvious direct tests of that hypothesis would be rather unethical, and the indirect tests (e.g. statistical correlations) would be vulnerable to selection biases. It's not just that schizophrenics might be more likely to disobey the law. There is apparently a part of the brain that is involved in religious experiences, e.g. Richard Dawkins had his stimulated and experienced nothing, apparently.
......If some people are more disposed to such things (again, whether they're born that way, or whether the brain adapts to their chosen way of life, is less obvious) then that would be a relatively obscure but effective source of such a bias. Incidentally, such studies need worry religious people surprisingly little. The traditional view of God has him timelessly creating us, and so faces similar problems, e.g. from our choices to turn to him etc. And Open-theistic views must face the facts of life anyway, e.g. that some of us are born richer, or better looking, or with better brains in other ways. If they can do that, they'll be able to live with similar facts.
......Anyway (oh how tangents attract the active mind), it occurs to me that those with such a religiously inclined brain might be more inclined (statistically, not each of them of course) to hear voices and also more inclined (similarly) to drink coffee, whether because they don't drink alcohol for religious reasons, or because they like to be awake to creation, or whatever. They may also be a little more inclined (statistically) to smoke cannabis, insofar as that's associated with the mystical side of hippies, or the religious side of Rastafarians, etc. If anything, we'd expect a greater corrolation with coffee, of course (and it does seem more plausible that coffee is not actually causing us to hear voices).
......Anyway, that's one possible explanation: a common cause leading to interlinked tendencies. Another explanation is that coffee in large quantities makes some of us irritable and tense, and perhaps over-sensitive, and so maybe more likely to hear voices insofar as we're already slightly inclined towards that (although I'm not ruling out the possibility that people take coffee because they're feeling stressed), but even in that possibility there's room for biases of the former kind—a common partial cause—e.g. a weak willed person might be more likely to over indulge in coffee, and less able to resist hallucinating. Similarly, they might drink more and get into fights, for such a reason. Or they might (also) react to the drink by getting more aggressive themselves. Note that that would be no reason for those who drink to relax and socialise to drink less (you may have guessed that I drink a lot of coffee:)

Faith, a definition

2009-01-13T10:15:00.002Z

Personal faith is not assent to evidence which is so strong as to be beyond reasonable doubt. It is assent to a discernment of God which is personally overwhelming but not objectively testable. This is not discernment of a historical God, timeless and unchanging. It is discernment of an active, loving God, making himself known in personal lives at specific points which become the matrix of a communal response to his will.

Keith Ward, Divine Action (London: Flame, 1990), 238.

How mysterious is Platonism?

2009-01-08T10:05:00.000Z

Arithmetical Platonism is supposed to be prima facie suspect because how, it’s asked, could we have arithmetical knowledge if the objects of that knowledge are in a world apart from us, a timeless world of Platonic objects, with which we cannot interact causally? I reply by wondering, how strange are abstract objects? You have just been reading this, for the obvious example. What have you been reading? You have been reading sentences. You look at the physical instances of these words, but you see the words, you read the words, and as you do so you are (hopefully) thinking about the thoughts expressed by means of them. So, there are, in the physical world around you, those physical instances of the shapes of (written modern English) words, and there are in your mind those thoughts; so, where are the sentences? What are the sentences? Sentences are made of words, and words are parts of a language (i.e. modern British English). They can be spoken or written, and can sometimes be spelt in different ways. Furthermore they have a meaning, a sense, and they have it essentially. The mere shape of a word is not a word, no more than meaningless strings of letters are words. Words, it seems, are abstract objects (I’m not entirely sure about that, or about what abstract objects are, so I’d welcome corrections) and you’ve just been reading some words of mine (and I’ll add that words can be true, insofar as they describe the world sufficiently accurately, or not, in case anyone wants to argue that thoughts and not words are truth-bearers:)

Who's the Philosopher?

2009-01-03T09:46:00.011Z

If good mathematics tells us that there are sets of some size, or functions of some type, or if good science tells us that there are waves or particles or forces or fields, then who is the philosopher to pipe up otherwise? As David Lewis forcefully argued, the philosopher taking any such line is apt only to make himself look foolish. Considering the case of sets in mathematics, Lewis wrote:
Mathematics is an established, going concern. Philosophy is as shaky as can be. To reject mathematics for philosophical reasons would be absurd [...] I’m moved to laughter at the thought of how presumptuous it would be to reject mathematics for philosophical reasons. How would you like the job of telling the mathematicians that they must change their ways, and abjure countless errors now that philosophy has discovered that there are no classes?
Philosophy simply has not got the track record of certainty, or utility, or progress, or unanimity, to mount any such high horse. If it is a question of philosophy versus physics, or philosophy versus maths, everyone knows which side to back.

That was Blackburn on Lewis on Philosophy, in Moore and Scott (eds.) Realism and Religion: Philosophical and Theological Perspectives (2007: 49), in the process of trying to draw a line between the expertise of scientists and that of theologians. This is a fairly common Naturalistic line, and there is, I think, a lot wrong with it.
......For a start, the Naturalists are usually pretty choosy about what they count as good science. And in justifying such choices they then try to draw a line between what scientists say about philosophical matters and what they say on scientific matters. As you might imagine, a lot of questions get begged in the process; a lot of the vanquished are straw men. But perhaps all that just goes to show how shaky philosophy is. So what I want to glance at here is what Lewis says about maths (to see how badly it fits with what Blackburn says about science), skipping over most of the fascinating nuances. After all, the argument is generally fallacious. E.g. a couple of years ago many would’ve said the same, as Lewis/Blackburn says about mathematics/physics and philosophy, about commerce and politics, and few think nowadays that markets did’t need some democratic regulation then.
......Mathematical classes necessarily go beyond standard set theory, as a result of the famous set-theoretic paradoxes of a hundred years ago. For the most part, their proper study is regarded by mathematicians and philosophers as the province of logicians and metaphysicians. And even the sets that mathematicians study are formal, axiomatic entities entirely suited to the algebraic ways of the mathematicians. Some mathematicians even work on constructive maths or category theory, for example, and insofar as they do so formally they are regarded as doing maths. In short, the pure mathematicians have passed the metaphysical buck to those applying the formal structures studied by them, while few applied mathematicians regard themselves as doing set theory.
......Lewis doesn’t accept that there are waves or particles or forces or fields, but only spacio-temporal points with arbitrary properties that either happen to fit a pattern or don’t. He’d let us keep whatever physicists say about such things, but would change their meanings (such being his idea of what philosophers can do). But those spacio-temporal points have, he presumes, a structure isomorphic to some set-theoretical structure. He might even allow that they could have any structure, but he needs some such points, and what if such structures happen to be metaphysically impossible? Standard mathematicians actually pass the buck on such applications.
......Lewis was talking about standard set theory, not mathematics (the two are often confused by American academics and those—most of the world’s leading academics—who’d like to become one), and sets have never got over the set-theoretical paradoxes. Or rather, the mathematicians—for the most part (Lewis conveniently ignores those mathematicians who’re constructivists or category theorists)—got round the problem by doing axiomatic set theory in an algebraic way, and thereby left the problem of interpretation up to those applying their maths. If there turn out to be no infinite sets really—no such metaphysically possible spatio-temporal points as Lewis presumes—then there would need to be some applicable mathematics done by someone, and whoever did it would be a mathematician. Lewis ought to be as sceptical about set theory as he is about particles and natural laws.
......Lewis may be moved to laughter at the very idea—a common response to his own hopelessly unrealistic work on metaphysics—but what was he thinking of? Brouwer and Heyting were professional mathematicians, and those philosophers (e.g. Wittgenstein and Dummett) who agreed with them were agreeing with mathematicians. So what was he thinking of, the maths that he was taught at school in America? If philosophers have something to contribute then they should contribute it, and if not then they’re philosophers in name only. Was Descartes a philosopher or a mathematician? He was clearly both, and we surely need more people like Descartes, not fewer. Scientists tend to say that to say nothing of God isn’t to say that God is nothing, and those who think otherwise do so for philosophical reasons.
......Mathematicians have as much right as anyone to think philosophically, and perhaps more right when it’s the philosophy of maths (e.g. Hamming and Fletcher). And what’s especially interesting nowadays is the increasing popularity of category theory, within maths. For the most part, that increasing popularity is not due to metaphysical concerns, but to concerns more internal to maths. But the underlying logic of category theory is intuitionistic. One can even envision a day when the professional mathematicians choose category theory as their standard foundation (one need only think of how popular Lewis is amongst professional philosophers) just because it provides the most interesting line of immediate research (cf. the dominance of string theory in physics)...
......et cetera ad nauseum (note that there’s some difference between philosophy the professional job (cf. Lewis’s concern with having the job of saying what some of his colleagues have said), which does seem shaky, and philosophy the pursuit of truth:)

Divine Liars: The devil's in the details

2009-01-02T09:42:00.004Z

Hartley Slater falsely supposed that I think there’s no essential difference between the sentential and the propositional formulations of (strengthened) Divine Liars, at the start of his ‘Supposed Liars, Divine Liars and Semantics’ (The Reasoner 3(1), 3–4). My ‘Liars, Divine Liars and Semantics’ (The Reasoner 2(12), 4–5) had merely questioned a presumption of Patrick Grim’s sententially formulated (4) = ‘God doesn’t believe that (4) is true.’ Grim had presumed without comment, let alone justification, that those two instances of ‘(4)’ could be the same proper name. And although such naming practices are fairly standard nowadays, that’s why the idea that their legitimacy is challenged by the paradoxicality of their application to Liar-style sentences could be of some interest.
......Propositional reformulations, such as my tentative (4*) = ‘God doesn’t believe that (4*) ever expresses a true proposition,’ raise different and more difficult considerations, which I’d simply ignored in order to state my question as clearly as I could within a thousand words. An obvious difference is that although a semantic problem with sentences is prima facie a problem with propositions, and vice versa, hardly anyone argues over whether sentences exist. Less obviously, although when considering (4)—which can be so-named even if I’m right about the illegitimacy of that naming practice, if the occurrence of ‘(4)’ within (4) isn’t also as (4)’s name—it had seemed acceptable to implicitly presuppose some ordinary linguistic notion of literal truth, perhaps I should make that notion more explicit when properly reformulating (4) propositionally. So instead of (4*) consider (4**) = ‘God doesn’t believe that the modern English sentence (4**) could ever express any true proposition literally.’
......That’s still pretty tentative, e.g. the modality might be difficult to explicate, and even literal truth is notoriously difficult to define. ‘Snow is white,’ for example, literally means that snow is white, and is true if indeed snow is white, but it’s hard to say what such repetitions mean in other, more general words. Still, there’s presumably some such semantic rule, say (T), that’s both true enough and applicable to such sentences as—for a simpler example (called ‘B’ earlier)—(L) = ‘This sentence is not (now) true.’ The parenthetical ‘now’ is only there to reduce the risk of equivocation, incidentally.
......Suppose, if only for the sake of reductio, that (L) was expressing something literally. What (L) would then be expressing would, via (T), at least include that (L) isn’t true, which is (in view of the self-reference) that it isn’t true that (L) isn’t true, which is (via double-negation elimination) that (L) is true. Now, what I’ve just shown is, I think, that the last two italicised expressions would be expressing the same proposition if (L) was being used to express any proposition literally. And maybe I’ve effectively shown that (L) expresses no proposition literally, i.e. that it’s nonsense. But I’ve certainly not thereby allowed that sentences may express more than one proposition. Recall that Slater said (ibid, 4):

Cooke says, with regard to ‘Liar sentences’, that ‘they do seem to be saying, not only that they are not true, but also, if less obviously, that they are (therefore) true’. So sentences, he allows, may express more than one proposition, even if they may express one proposition more obviously than another. But if so then one cannot immediately derive, with respect to the previous case that the (one and only) proposition that (4*) expresses is (the obvious one) that God doesn’t believe that (4*) ever expresses a true proposition.

A sentence may of course express different propositions, e.g. literally and analogically, or by being equivocal, or when it’s expressed by different people, or at different times or places, or because the language in which it exists changes, etc. But Slater will, I suspect, have difficulty indicating what other proposition could have been expressed by (4*)—or better, (4**)—literally. If he has to use different words to those of (4*), then is it really expressed by (4*)? And if he doesn’t, then why wasn’t it expressed when he used those same words to express the ‘obvious’ proposition?
......Furthermore even if one may, in such a way as Slater’s (above), point unambiguously to one proposition, something like the original problem with (4) would arise. Note that if (4**)—or (4*)—is, as I believe it is, nonsense, then it doesn’t express any proposition literally, and so ideally one wouldn’t be able to derive that it does. But if, counterfactually, (4**) was expressing anything literally, then presumably by (T) that would be the proposition that God doesn’t believe that the modern English sentence (4**) could ever express any true proposition literally. And that does seem like a proposition to me, if only because I’d assert it, in so many words, on the grounds that (4**) is literal nonsense and that God would, were he real, be wise enough to know that. So, whether or not any other propositions are expressible by (4**), literally or otherwise, we’ve something like the problem that (4) presented to theists. That is, if (4**) could be used to express a true thought literally then God doesn’t believe—via the truth of that (last italicised) proposition—that it could be so used, whereas if (4**) couldn’t be so used then that proposition is true, whence any completely omniscient being would believe it.
......Something like it; but as I say, it may be that a reformulation more precise than (4**) is required to yield a paradox sufficiently close to that of (4). In fact, I happen to be agnostic about whether that’s even possible, about whether there’s at least that essential difference between the two sorts of formulation. But the point is that the traditional resolution of Liar-style sentences—that you can’t, by talking nonsense, say that you aren’t telling the truth—applies to any legitimate formulation, and would in particular apply to sentential formulations, were they allowed.

The Pope and the Archbishop

2008-12-23T10:07:00.010Z

I just noticed, as I wrote that last post, the Pope on the TV saying more stuff against homosexuality, which is best seen in the context of what the Archbishop of Canterbury was saying a few days ago, about borrowing our way out of a mess that we got into via greedy borrowing being a bit like what alcoholics do. That mess was, I think, more to do with who the City employed and why; but still, why did we let them get away with it? And why'll we probably do the same again? The thing about homosexuality is that it's very emotive. I'm sure the Pope isn't unaware that he's pandering to homophobia, even if he talks about the sin and not the sinner. After all, a similar sin is thinking lustful thoughts about a woman other than one's wife, e.g. when watching a movie (which a lot will be doing this xmas). Marriages tend to break up because they are felt to be falling short of the dream, not because gays come out of closets. And a worse sin is pride, of course; especially pride in such trivia as being straight, or white. The thing about homophobia is that it's obviously like racism and antisemitism etc. People, even straight people, often feel insecure about their sexuality (as the Church has traditionally encouraged them to), and a way to feel better about it with very little subjectively obvious psychic cost (if one's straight) is to think to yourself that at least you're not one of those disgusting gays. The similarities with racism and poor people's views of their own social positions are obvious (not to mention the Church's traditional role in antisemitism). A man can know that he's sexist and letcherous, but comfort himself with the thought that at least he fancies women. And a woman can know that she's fat and lazy, but comfort herself with the thought that at least she's a woman. The irony is that the Church traditionally regarded marriage as second-best to the monastic life (and another irony is that the latter attracted homosexuals, of course... I could go on, but I'll just wish you a merry xmas :)

A Jump Theodicy

2008-12-23T09:38:00.010Z

Another name-change for my theodicy, to something less irritatingly alliterative and more evocative of its content (by comparison with Fall theodicies), which is now here (Alanyzer's thoughts on it are here). Last January my theodicy was a sketchy talk entitled ‘The Theodolite Theodicy’ at Glasgow (where I was supposed to be continuing with the metaphysics of continuity for my PhD), at which it matured slightly under the questions of Fiona and Akiko.
......In the Spring I happened upon Tim Mawson’s recent paper in Int. J. Philos. Relig. (which was essentially the second chapter of his 2005 introduction to the philosophy of religion), and the theodicy—renamed the Odyssey theodicy—became the final part of my refutation of his arguments for the timelessness of God. I’d finished writing a response to Mawson’s paper (which was essentially this version) by the time of my talk in Aberdeen in July; and its rejection arrived in November, along with three reasons for its rejection, which were so weak as to be interesting.
......The first objection was to my use of the term ‘theodicy’ (as opposed to ‘defence’) on the grounds that, while the reviewer conceded that my speculations might be possible, she (or he) didn’t find them at all plausible. But when atheists find no (so-called) theodicy plausible, are all theodicies thereby misnamed? Hardly, and so (similarly) that she found my theodicy implausible hardly stops it being a theodicy. It should’ve been obvious that I’m not trying to demonstrate the logical compatibility of God’s existence and evil’s occurrence (which is surely trivial) but to maximise the explanatory power of Open theism, by trying to give a good account of why a perfect being would make an imperfect world.
......So her first objection amounted—at best (it may just have been incompetent, in view of the quality of the other two)—to no more than the claim that I’d failed to give a good account of that. As for why I had so failed, there were only the following two objections. Since philosophy ought to be more like amateur science than a professional game, I’d rather add that had she been able to ask me, I could’ve easily cleared up her confusions. Such is blind reviewing.
......Her second objection was that, while one of the aims of Open theism is to bring the philosophical picture of God closer to the Biblical picture, my theodicy would forfeit that aim. However, she said nothing about why it would. And having read the Bible inclusivistically (e.g. with metaphysical humility) and found no incompatibility with my theodicy, I don’t know which verses she was thinking of (or how). If the readers of this post have any ideas of what they might be, I’d be very interested in any possibilities. My theodicy could hardly take us further from the Biblical picture(s) than the doctrine of God’s timelessness has traditionally taken us.
......But what’s most apposite, from the point of view of reviewing a submission, is that even were this objection sound the first half of my submission would still have shown that a perfect person might be everlasting (contra Mawson) and indeed, would be (according to Mawson’s own methodology), while the final half would still have further increased the likelihood of Open theism.
......The final objection was basically a Straw Man fallacy, and was (in full) as follows.

There are several arguments in the literature that it is not possible that there be two omnipotent beings. The relevance of these arguments to the author’s project is obvious. But the soundness of these arguments is no where contested in the paper. If these arguments are sound, then God, as omnipotent, can be quite confident that there are no other unknown deities about.

God presumably is omnipotent but, as I’d argued, it hardly follows that he could be fully justified in being completely sure that he is. And clearly, if God is only fairly confident (and fully justified in being so) then there is, for him, the epistemic possibility that grounds my theodicy. None of those arguments of mine were criticised by her, as though she was unaware of them (despite their obvious relevance). But a trivial consequence of them is that the arguments she mentioned are none of them relevant (not even the one published alongside Mawson).

What Next? Turkey

2008-12-17T14:06:00.006Z

For Turkey to become an EU member would change the Union, a point that we should not seek to hide from our citizens. The European demos has to be party to the deal, and it should not be impossible to convince the public of the Turkish case if we try. A Turkey with a population of rising 90 million with a vibrant economy and young workforce would help to vitalize Europe’s economy as the west-European population ages and declines. A democratic Turkey, secular and Muslim, would assist in preventing the cultural divisions and clashes that we sometimes appear intent on provoking around the world. A militarily professional Turkey would give the EU more credibility as a civilian power able to act occasionally with an effective military smack. Reject Turkey and the EU will have chosen to write a much smaller part for itself in the history of the twenty-first century, and having been so successful in promoting stability around our borders we may find ourselves doing the reverse.

That’s from Lord Patten’s ‘What Next? Surviving the Twenty-first Century’ (2008: Allen Lane, p. 419), which everyone should read, if only to get clear on what’s been happening recently.

The Dark Compliment

2008-11-25T10:42:00.011Z

I’m reading about the philosophy of physics (see previous post), so I happen to be thinking of Bohr's wave-particle complimentarity... and resonance:
......An empty bottle, for example, emits a low tone when you blow over it because the tiny vibrations that are the disturbance of the air caused by that blowing all mount up if (and only if) they have a wavelength that fits nicely into the extent of the bottle (much as how, when you give a swing a sequence of little pushes, the swing swings with a larger and larger amplitude). I’m thinking that because the universe has a finite size, so there might be resonance. Maybe the universe is still ringing like a bell from the Big Bang. It is clearly permeated by background radiation (the lingering whisper of that explosion), so maybe there is also resonance. (The universe is growing, so the resonant notes would be lowering their tone.)
......Where is the resonant ringing? Well if the resonance was like a sound wave, we would expect to see bands of denser particles and less dense particles (that being what sound is), those particles being galaxies perhaps, or clusters of galaxies. And if you think of the waves being in spacetime then again, matter would tend to congregate in the troughs. Small ripples move over ocean swells much as they do over calm seas, so we would not necessarily notice anything locally; but such clustering as though in troughs is indeed observed at the largest scales. And most of the wave energy is in those huge swells, which reminds me of the dark matter that, whilst being unobservable (whence the term 'dark'), is thought to make up over 80% of the matter of the universe.
......Galaxies seem to need more mass than we can see, in their stars and dust, to account for how compacted together are those stars and that dust. So my thought is that dark matter might be the energy associated with such huge universal swells. It would be directly unobservable as matter because such swells would be too big to look much like particles to us, much as electrons are too small to look like waves (except when that aspect proves invaluable, e.g. in electron microscopy). Maybe not of course, but I was wondering if any reader knows whether or not the maths of that analogy works out?

Thinking things through thoroughly

2008-11-24T14:08:00.009Z

An atom with stationary electrons positioned around a positive nucleus would be unstable, because the electrons with their negative charge would be irresistibly pulled towards it. If they moved around the nucleus, like planets orbiting the sun, the atom would still collapse. Newton had shown long ago that any object moving in a circle undergoes acceleration. According to Maxwell’s theory of electromagnetism, if it is a charged particle, like an electron, it will continuously lose energy in the form of electromagnetic radiation as it accelerates. An orbiting electron would spiral into the nucleus within a thousandth of a billionth of a second. The very existence of the material world was compelling evidence against Rutherford’s nuclear atom.

That's (p. 81) from Quantum, an exceptionally good account of the revolution in physics in the first half of the last century (the physics is nicely explained (the author studied philosophy as well as physics) and most importantly, the biography is as engaging as in a novel (if a literary one))... I quoted the paragraph above because of its combination of logical clarity, scientific support and falsity. The resolution begins to emerge in 1913 (p. 96):

Whereas others had interpreted these problems of instability as damning evidence against Rutherford’s nuclear atom, for Bohr they signalled the limitations of the underlying physics that predicted its demise. His identification of radioactivity as a ‘nuclear’ and not an ‘atomic’ phenomenon, his pioneering work on radioelements, what Soddy later called isotopes, and on nuclear charge convinced Bohr that Rutherford’s atom was indeed stable. Although it could not bear the weight of established physics, it did not suffer the predicted collapse. The question that Bohr had to answer was: why not?

Bohr’s electron shell model of the atom was ready to explain chemistry by 1922, and in 1923 it was physically justified by a French prince who had earlier failed his physics exams (p. 149):

If viewed as a standing wave around the nucleus instead of a particle in orbit, an electron would experience no acceleration and therefore no continual loss of radiation sending it crashing into the nucleus as the atom collapsed. What Bohr had introduced simply to save his quantum atom, found its justification in de Broglie’s wave-particle duality. When he did the calculations, de Broglie found that Bohr’s principal quantum number, n, labelled only those orbits in which electron standing waves could exist around the nucleus of the hydrogen atom. It was the reason why all other electron orbits were forbidden in the Bohr model.

In late 1925 Schrodinger read of de Broglie’s idea in a footnote to one of Einstein’s papers, and by early 1926 he had obtained his famous wave equation (p. 206):

Schrodinger knew exactly where to start and what he had to do. De Broglie had tested his idea of wave-particle duality by reproducing the allowed electron orbits in the Bohr atom as those in which only a whole number of standing electron wavelengths could fit. Schrodinger knew that the elusive wave equation he sought would have to reproduce the three-dimensional model of the hydrogen atom with three-dimensional standing waves. The hydrogen atom would be the litmus test for the wave equation he needed to find.

Schrodinger disagreed with the probabilistic interpretation of his wave equation introduced by Max Born that same year; but Bohr was more perceptive (p. 219):

Niels Bohr would soon argue that until an observation or measurement is made, a microphysical object like an electron does not exist anywhere. Between one measurement and the next it has no existence outside the abstract possibilities of the wave function. It is only when an observation or measurement is made that the ‘wave function collapses’ as one of the ‘possible states of the electron becomes the ‘actual’ state and the probability of all the other possibilities becomes zero.

Richards' paradox again

2008-11-21T16:16:00.002Z

Upon reflection, I don’t think much of my previous resolution of Richards’ Paradox, which was as follows: Such finite strings of words as specify real numbers between 0 and 1 can be listed in order of increasing number of symbols used in the description and then, within each length of description, in alphabetical order. Given that list, one may seem able to define a real number between 0 and 1 and not in that list using a diagonal procedure: The digit in its nth decimal place is the final digit of d + 1, where d is the digit in the nth decimal place of the nth number on the list. Richards’ paradox is that such a number has such a finite description (so it is in our list, whence that procedure is contradictory, where it is, so there is no such number, in our list, whence that procedure is not contradictory, and so on).
......My previous resolution was just to note that our list includes anything that anyone could possible say (in finitely many words) that would specify a real number between 0 and 1. So in order to specify a number via that list, any (finitely describable) diagonal procedure must explicitly exclude its own entries in the list. But that now seems patently inadequate. Given such a list, we have its diagonal, and so a real number not on that list is clearly indicated. And we can say that it is. What we say cannot be in the list, but that just means that such a list—of all the finite specifications of such real numbers—is impossible. I would not be surprised if it was impossible, because words are typically a bit vague and variable in meaning. But it does seem odd that we can deduce that they must be fuzzy or incomplete, which is why I am not very fond of that resolution either. All in all, I find Richards’ paradox very puzzling.

How We Reason

2008-11-17T15:00:00.009Z

*......Either Jane is kneeling by the fire and she is looking at the TV, or else Mark is standing at the window and he is peering into the garden.

*......Jane is kneeling by the fire.
Does it follow that she is looking at the TV?
......Most individuals say, “yes”, see Walsh, C., and Johnson-Laird, P.N. (2004: Co-reference and reasoning. Memory & Cognition, 32, 96–106). Given the first premise, they think of two possibilities: in one, the first conjunction is true; and in the other, the second conjunction is true. They overlook that when the second conjunction is true, the first conjunction is false, and that one way in which it can be false is when only its first clause is true, i.e., Jane is kneeling by the fire but not looking at the TV. Hence, the correct answer to the question is: “no”.

That’s from ‘How We Reason: A view from Psychology’ in The Reasoner 2(3), 4–5. Philip Johnson-Laird’s book ‘How We Reason’ is out in paperback next month.
......Maybe that was an example of a fallacy, rather than a paradox, but I’ve labelled it under ‘paradox’ in view of how easy it is to see through those I’ve been looking at recently; and after all, I’d said “yes” myself. (There’s a nice list of fallacies in The Reasoner 2(5), 7–8.)